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Enumerating conjugacy classes of graphical groups over finite fields

Enumerating conjugacy classes of graphical groups over finite fields INTRODUCTIONGraphical groupsThroughout, graphs are finite, simple, and (unless otherwise indicated) contain at least one vertex. When the reference to an ambient graph is clear, we use ∼$\sim$ to indicate the associated adjacency relation. All rings are associative, commutative, and unital.Let Γ=(V,E)$\Gamma = (V,E)$ be a graph with n$n$ vertices. The graphical group GΓ(R)$\mathbf {G}_{\Gamma} (R)$ associated with Γ$\Gamma$ over a ring R$R$ was defined in [29, Section 3.4]. For a short equivalent description (see Section 2.3), write V={v1,…,vn}$V = \lbrace v_1,\ldots\,,v_n\rbrace$ and let J={(j,k):1⩽j<k⩽n,vj∼vk}$J = \lbrace (j,k) : 1\leqslant j &lt; k \leqslant n,\, v_j\sim v_k\rbrace$. Then GΓ(R)$\mathbf {G}_{\Gamma} (R)$ is generated by symbols x1(r),…,xn(r)$x_1(r),\ldots\,,x_n(r)$ and zjk(r)$z_{jk}(r)$ for (j,k)∈J$(j,k)\in J$ and r∈R$r\in R$, subject to the following defining relations for i,i′∈[n]:={1,…,n}$i,i^{\prime } \in [n] := \lbrace 1,\ldots\,,n\rbrace$, (j,k),(j′,k′)∈J$(j,k), (j^{\prime },k^{\prime })\in J$, and r,r′∈R$r,r^{\prime }\in R$:(i)xi(r)xi(r′)=xi(r+r′)$x_i(r) x_i(r^{\prime }) = x_i(r+r^{\prime })$ and zjk(r)zjk(r′)=zjk(r+r′)$z_{jk}(r)z_{jk}(r^{\prime }) = z_{jk}(r+r^{\prime })$.       (‘scalars')(ii)[xj(r),xk(r′)]=zjk(rr′)$[x_j(r),x_k(r^{\prime })] = z_{jk}(rr^{\prime })$.       (‘adjacent vertices and commutators')(Recall that (j,k)∈J$(j,k) \in J$ so that vj∼vk$v_j \sim v_k$.)(iii)[xi(r),xi′(r′)]=1$[x_i(r),x_{i^{\prime }}(r^{\prime })] = 1$ if vi≁vi′$v_i\notsim v_{i^{\prime }}$.       (‘non‐adjacent vertices and commutators')(iv)[xi(r),zjk(r′)]=[zj′k′(r),zjk(r′)]=1$[x_i(r), z_{jk}(r^{\prime })] = [ z_{j^{\prime }k^{\prime }}(r),z_{jk}(r^{\prime })]= 1$.       (‘centrality of commutators')Note that every ring map R→R′$R\rightarrow R^{\prime }$ induces an evident group homomorphism GΓ(R)→GΓ(R′)$\mathbf {G}_{\Gamma} (R) \rightarrow \mathbf {G}_{\Gamma} (R^{\prime })$. We will see in Section 2.3 that the resulting group functor GΓ$\mathbf {G}_{\Gamma}$ represents the graphical group scheme associated with Γ$\Gamma$ as defined in [29]. The isomorphism type of GΓ$\mathbf {G}_{\Gamma}$ does not depend on the chosen ordering of the vertices of Γ$\Gamma$.1.1ExampleVarious instances and relatives of graphical groups appeared in the literature.(i)GΓ(Z)$\mathbf {G}_{\Gamma} (\mathbf {Z})$ is isomorphic to the maximal nilpotent quotient of class at most 2 of the right‐angled Artin group ⟨x1,…,xn∣[xi,xj]=1whenevervi≁vj⟩$\langle x_1,\ldots\,,x_n \mid [x_i,x_j] = 1 \text{ whenever } v_i\notsim v_j\rangle$; see Section 2.1.(ii)Let Kn$\operatorname{K}_n$ denote a complete graph on n$n$ vertices. Then GKn(Z)$\mathbf {G}_{\operatorname{K}_n}(\mathbf {Z})$ is a free nilpotent group of rank n$n$ and class at most 2. For each odd prime p$p$, the graphical group GKn(Fp)$\mathbf {G}_{\operatorname{K}_n}(\mathbf {F}_p)$ is a free nilpotent group of rank n$n$, exponent dividing p$p$, and class at most 2. (Both statements follow from Proposition 2.1 below. We note that the prime 2 does not play an exceptional role in any of our main results.)(iii)Let Pn$\operatorname{P}_{{n}}$ be a path graph on n$n$ vertices. Let Ud⩽GLd$\operatorname{U}_d\leqslant \operatorname{GL}_d$ be the group scheme of upper unitriangular d×d$d\times d$ matrices. Then for each ring R$R$, the group GPn(R)$\mathbf {G}_{\operatorname{P}_{{n}}}(R)$ is the maximal quotient of class at most 2 of Un+1(R)$\operatorname{U}_{n+1}(R)$; cf. [29, Section 9.4].(iv)Let Δn$\Delta _n$ denote an edgeless graph on n$n$ vertices. Then for each ring R$R$, we may identify GΔn(R)$\mathbf {G}_{\Delta _n}(R)$ and the (abelian) additive group Rn$R^n$.(v)Let p$p$ be an odd prime. Then GΓ(Fp)$\mathbf {G}_{\Gamma} (\mathbf {F}_p)$ is isomorphic to the p$p$‐group attached to the complement of Γ$\Gamma$ via Mekler's construction [22]; cf. Proposition 2.1(ii). Li and Qiao [17] used what they dubbed the Baer‐Lovász–Tutte procedure to attach a finite p$p$‐group to Γ$\Gamma$. Their group is also isomorphic to GΓ(Fp)$\mathbf {G}_{\Gamma} (\mathbf {F}_p)$; see Section 2.4.Known results: Class numbers of graphical groupsLet cce(G)${\rm cc}_e(G)$ denote the number of conjugacy classes of size e$e$ of a finite group G$G$, and let k(G)=∑e=1∞cce(G)$\operatorname{k}(G) = \sum _{e=1}^\infty {\rm cc}_e(G)$ be the class number of G$G$. It is well known that k(GLd(Fq))$\operatorname{k}(\operatorname{GL}_d(\mathbf {F}_q))$ is a polynomial in q$q$ for fixed d$d$; see [31, Chapter 1, Exercise 190]. This article is devoted to the class numbers k(GΓ(Fq))$\operatorname{k}(\mathbf {G}_{\Gamma} (\mathbf {F}_q))$. We first recall known results.1.2Theorem[29, Cor. 1.3]Given a graph Γ$\Gamma$, there exists fΓ(X)∈Q[X]$f_{\Gamma} (X)\in \mathbf {Q}[X]$ such that k(GΓ(Fq))=fΓ(q)$\operatorname{k}(\mathbf {G}_{\Gamma} (\mathbf {F}_q)) = f_{\Gamma} (q)$ for each prime power q$q$.We call fΓ(X)$f_{\Gamma} (X)$ the class‐counting polynomial of Γ$\Gamma$. In [29], Theorem 1.2 is derived from a more general uniformity result [29, Cor. B] for class‐counting zeta functions associated with graphical group schemes; see Section 8.1. Formulae for fΓ(X)$f_{\Gamma} (X)$ when Γ$\Gamma$ has at most 5 vertices can be deduced from the tables in [29, Section 9]. Moreover, several families of class‐counting polynomials have been previously computed in the literature.1.3ExampleO'Brien and Voll [24, Thm 2.6] gave a formula for the number of conjugacy classes of given size of p$p$‐groups derived from free nilpotent Lie algebras via the Lazard correspondence. Using the interpretation of GKn(Fp)$\mathbf {G}_{\operatorname{K}_n}(\mathbf {F}_p)$ in Example 1.1(ii), their formula or, alternatively, work of Ito and Mann [16, Section 1] yields fKn(X)=Xn−12(Xn+Xn−1−1)$f_{\operatorname{K}_n}(X) = X^{\binom{n-1}{2}}(X^n + X^{n-1}-1)$.1.4ExampleIn light of Example 1.1(iii), Marjoram's enumeration [21, Thm 7] of the irreducible characters of given degree of the maximal class‐2 quotients of Ud(Fq)$\operatorname{U}_d(\mathbf {F}_q)$ yields1.1fPn(X)=∑a=0n2n−aaXn−a−1(X−1)a+n−a−1aXn−a−1(X−1)a+1.\begin{equation} f_{\operatorname{P}_{{n}}}(X) = \sum\limits _{a=0}^{\left\lfloor \frac{n}{2}\right\rfloor } {\left(\binom{n-a}{a} X^{n-a-1}(X-1)^a + \binom{n-a-1}{a}X^{n-a-1}(X-1)^{a+1}\right)}. \end{equation}For any graph Γ$\Gamma$, the size of each conjugacy class of GΓ(Fq)$\mathbf {G}_{\Gamma} (\mathbf {F}_q)$ is of the form qi$q^i$; see Proposition 2.3(i). As indicated in [29, Section 8.5], the methods underpinning Theorem 1.2 can be used to strengthen said theorem: each ccqi(GΓ(Fq))${\rm cc}_{q^i}(\mathbf {G}_{\Gamma} (\mathbf {F}_q))$ is a polynomial in q$q$ with rational coefficients. While constructive, the proof of Theorem 1.2 in [29] relies on an elaborate recursion. In particular, no explicit general formulae for the numbers ccqi(GΓ(Fq))${\rm cc}_{q^i}(\mathbf {G}_{\Gamma} (\mathbf {F}_q))$ or the polynomial fΓ(X)$f_{\Gamma} (X)$ have been previously recorded.Recall that the join Γ1∨Γ2$\Gamma _1\vee \Gamma _2$ of graphs Γ1$\Gamma _1$ and Γ2$\Gamma _2$ is obtained from their disjoint union Γ1⊕Γ2$\Gamma _1\oplus \Gamma _2$ by adding edges connecting each vertex of Γ1$\Gamma _1$ to each vertex of Γ2$\Gamma _2$. Further recall that a cograph is any graph that can be obtained from two cographs on fewer vertices by taking disjoint unions or joins, starting with an isolated vertex.1.5TheoremCf. [29, Theorem E]Let Γ$\Gamma$ be a cograph. Then the coefficients of fΓ(X)$f_{\Gamma} (X)$ as a polynomial in X−1$X-1$ are non‐negative integers.Theorems 1.2 and 1.5 are special cases of more general results pertaining to class numbers of graphical groups GΓ(O/Pi)$\mathbf {G}_{\Gamma} (\mathfrak {O}/\mathfrak {P}^i)$, where O$\mathfrak {O}$ is a compact discrete valuation ring with maximal ideal P$\mathfrak {P}$. We will briefly discuss this topic in Section 8.The graph polynomials CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ and FΓ(X,Y)$\mathsf {F}_{\Gamma} (X,Y)$As before, let Γ=(V,E)$\Gamma = (V,E)$ be a graph. Prior to stating our results, we first define what, to the author's knowledge, appears to be a new graph polynomial. For a (not necessarily proper) subset U⊂V$U\subset V$, let Γ[U]$\Gamma [U]$ be the induced subgraph of Γ$\Gamma$ with vertex set U$U$. Let cΓ(U)$\operatorname{c}_{\Gamma} (U)$ denote the number of connected components of Γ[U]$\Gamma [U]$. (We allow U=∅$U = \emptyset$ in which case cΓ(U)=0$\operatorname{c}_{\Gamma} (U) = 0$.) The closed neighbourhood NΓ[v]⊂V$\operatorname{N}_{\Gamma} [v] \subset V$ of v∈V$v\in V$ consists of v$v$ and all vertices adjacent to it. For U⊂V$U\subset V$, write NΓ[U]=⋃u∈UNΓ[u]$\operatorname{N}_{\Gamma} [U] = \bigcup _{u\in U}\operatorname{N}_{\Gamma} [u]$. Define1.2CΓ(X,Y)=∑U⊂V(X−1)|U|Y|NΓ[U]|−cΓ(U)∈Z[X,Y].\begin{equation} \mathcal {C}_{\Gamma} (X,Y) = \sum _{U\subset V} (X-1)^{\vert U\vert }\, Y^{\vert \operatorname{N}_{\Gamma} [U]\vert - \operatorname{c}_{\Gamma} (U)} \in \mathbf {Z}[X,Y]. \end{equation}1.6RemarkWhile CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ resembles the Tutte polynomial of a matroid on the ground set V$V$, it is unclear to the author whether this is more than a formal similarity. Similarly, CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ is reminiscent of the subgraph polynomial [26, Section 3] of Γ$\Gamma$.Let Γ$\Gamma$ have n$n$ vertices, m$m$ edges, and c$c$ connected components. Recall that the (matroid) rank of Γ$\Gamma$ is rk(Γ)=n−c$\operatorname{rk}(\Gamma ) = n-c$. Define the class‐size polynomial of Γ$\Gamma$ to be1.3FΓ(X,Y)=XmCΓ(X,X−1Y)=∑U⊂V(X−1)|U|Xm+cΓ(U)−|NΓ[U]|Y|NΓ[U]|−cΓ(U).\begin{equation} \mathsf {F}_{\Gamma} (X,Y) = X^m \mathcal {C}_{\Gamma} (X,X^{-1}Y) = \sum _{U\subset V} (X-1)^{\vert U\vert } X^{m + \operatorname{c}_{\Gamma} (U) -\vert \operatorname{N}_{\Gamma} [U]\vert } Y^{\vert \operatorname{N}_{\Gamma} [U]\vert -\operatorname{c}_{\Gamma} (U)}. \end{equation}The degree of CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ as a polynomial in Y$Y$ is rk(Γ)$\operatorname{rk}(\Gamma )$; see Proposition 6.2. As rk(Γ)⩽m$\operatorname{rk}(\Gamma ) \leqslant m$, we conclude that FΓ(X,Y)∈Z[X,Y]$\mathsf {F}_{\Gamma} (X,Y) \in \mathbf {Z}[X,Y]$. While CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ and FΓ(X,Y)$\mathsf {F}_{\Gamma} (X,Y)$ determine each other, CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ is often more convenient to work with and FΓ(X,Y)$\mathsf {F}_{\Gamma} (X,Y)$ turns out to be more directly related to the enumeration of conjugacy classes; see Theorem A.1.7Example   (i)CKn(X,Y)=(Xn−1)Yn−1+1$\mathcal {C}_{\operatorname{K}_n}(X,Y) = (X^n -1) Y^{n-1} + 1$ and FKn(X,Y)=Xn2+1−Xn−12Yn−1+Xn2$\mathsf {F}_{\operatorname{K}_n}(X,Y) = \Bigl (X^{\binom{n}{2}+1}-X^{\binom{n-1}{2}}\Bigr )Y^{n-1} + X^{\binom{n}{2}}$.(ii)CΔn(X,Y)=Xn=FΔn(X,Y)$\mathcal {C}_{\Delta _n}(X,Y) = X^n = \mathsf {F}_{\Delta _n}(X,Y)$.Main resultsLet Γ$\Gamma$ be a graph. The main result of this article justifies the term ‘class‐size polynomial’.ATheoremFΓ(q,Y)=∑i=0∞ccqi(GΓ(Fq))Yi$\mathsf {F}_{\Gamma} (q,Y) = \sum _{i=0}^\infty {\rm cc}_{q^i}(\mathbf {G}_{\Gamma} (\mathbf {F}_q)) Y^i$ for each prime power q$q$.Note that ccqi(GΓ(Fq))=0${\rm cc}_{q^i}(\mathbf {G}_{\Gamma} (\mathbf {F}_q)) = 0$ for all sufficiently large i$i$ so Theorem A asserts an equality of polynomials in Y$Y$.1.8ExampleThe formula in [24, Thm 2.6] referred to in Example 1.3 shows that if q$q$ is an odd prime power, then GKn(Fq)$\mathbf {G}_{\operatorname{K}_n}(\mathbf {F}_q)$ has a centre of order qn2$q^{\binom{n}{2}}$ and precisely (qn−1)qn−12$(q^n-1)q^{\binom{n-1}{2}}$ non‐trivial conjugacy classes, all of size qn−1$q^{n-1}$. These numbers agree with Example 1.7(i). Clearly, Example 1.7(ii) agrees with the fact that GΔn(Fq)≈Fqn$\mathbf {G}_{\Delta _n}(\mathbf {F}_q) \approx \mathbf {F}_q^n$ is abelian.Theorem A provides us with the following explicit formula for the class‐counting polynomial fΓ(X)$f_{\Gamma} (X)$ defined in Theorem 1.2:BCorollaryfΓ(X)=FΓ(X,1)=∑U⊂V(X−1)|U|Xm+cΓ(U)−|NΓ[U]|$ f_{\Gamma} (X) = \mathsf {F}_{\Gamma} (X,1) = \sum _{U\subset V} (X-1)^{\vert U\vert }\, X^{m+\operatorname{c}_{\Gamma} (U)-\vert \operatorname{N}_{\Gamma} [U]\vert }$.Note that Corollary B shows that fΓ(X)$f_{\Gamma} (X)$ has integer coefficients. In the spirit of work surrounding Higman's conjecture (see Section 1.5) and Theorem 1.5, Theorem A implies the following refinement of the preceding observation:CCorollaryFor each e⩾1$e\geqslant 1$, the number of conjugacy classes of GΓ(Fq)$\mathbf {G}_{\Gamma} (\mathbf {F}_q)$ of size e$e$ is given by a polynomial in q−1$q-1$ with non‐negative integer coefficients.ProofUse the binomial theorem to expand powers of X=(X−1)+1$X = (X-1) + 1$ in (1.3).□$\Box$It is natural to ask whether the coefficients referred to in Corollary C enumerate meaningful combinatorial objects. Corollary 6.5 will provide a partial answer to this.We shall not endeavour to improve substantially upon the exponential‐time algorithm for computing CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ suggested by Equation (1.2). Indeed, we will obtain the following:DPropositionComputing CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$, and hence also FΓ(X,Y)$\mathsf {F}_{\Gamma} (X,Y)$, is NP‐hard.More precisely, we will see that knowledge of CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ allows us to read off the cardinalities of connected dominating sets of Γ$\Gamma$. The problem of deciding whether a graph admits a connected dominating set of cardinality at most a given number is known to be NP‐complete; see Theorem 6.6.By Theorem A and Proposition D, symbolically enumerating the conjugacy classes of given size of GΓ(Fq)$\mathbf {G}_{\Gamma} (\mathbf {F}_q)$ (as a polynomial in q$q$) is NP‐hard. The problem of measuring the difficulty of symbolically enumerating all conjugacy classes of GΓ(Fq)$\mathbf {G}_{\Gamma} (\mathbf {F}_q)$ remains open.1.9QuestionIs computing fΓ(X)$f_{\Gamma} (X)$ NP‐hard?Related work: Around Higman's conjectureRecall that Ud⩽GLd$\operatorname{U}_d\leqslant \operatorname{GL}_d$ denotes the group scheme of upper unitriangular d×d$d\times d$ matrices. A famous conjecture due to G. Higman [14] predicts that k(Ud(Fq))$\operatorname{k}(\operatorname{U}_d(\mathbf {F}_q))$ is given by a polynomial in q$q$ for fixed d$d$. This has been confirmed for d⩽13$d\leqslant 13$ by Vera‐López and Arregi [35] and for d⩽16$d\leqslant 16$ by Pak and Soffer [25]. The former authors also showed that the sizes of conjugacy classes of Ud(Fq)$\operatorname{U}_d(\mathbf {F}_q)$ are of the form qi$q^i$ (see [36, Section 3]) and that ccqi(Ud(Fq))${\rm cc}_{q^i}(\operatorname{U}_d(\mathbf {F}_q))$ is a polynomial in q−1$q-1$ with non‐negative integer coefficients for i⩽d−3$i \leqslant d-3$ (see [34]). Many authors studied variants of Higman's conjecture for unipotent groups derived from various types of algebraic groups; see, for example, [10].While logically independent of the work described here, Higman's conjecture (and the body of research surrounding it) certainly provided motivation for topics considered and results obtained in this article (for example, Corollary C).Open problems: Enumerating characters of graphical groupsLet Irr(G)$\operatorname{Irr}(G)$ denote the set of (ordinary) irreducible characters of a finite group G$G$. It is well known that k(G)=|Irr(G)|$\operatorname{k}(G) = \vert \operatorname{Irr}(G)\vert$ (see, for example,[15, V, Section 5]), and the enumeration of irreducible characters of a group (according to their degrees) has often been studied as a ‘dual’ of the enumeration of conjugacy classes (according to their sizes); see, for example, [18, 24, 28]. For odd q$q$, [24, Thm B] implies that the degree χ(1)$\chi (1)$ of each irreducible character χ$\chi$ of a graphical group GΓ(Fq)$\mathbf {G}_{\Gamma} (\mathbf {F}_q)$ is of the form qi$q^i$.1.10QuestionLet Γ$\Gamma$ be a graph and let i⩾0$i\geqslant 0$ be an integer. How doesch(Γ,i;q):=#χ∈Irr(GΓ(Fq)):χ(1)=qi\begin{equation*} \mathrm{ch}(\Gamma ,i;q) := \#{\left\lbrace \chi \in \operatorname{Irr}(\mathbf {G}_{\Gamma} (\mathbf {F}_q)) : \chi (1) = q^i\right\rbrace} \end{equation*}depend on the prime power q$q$?It is known that ch(Γ,i;q)$\mathrm{ch}(\Gamma ,i;q)$ is a polynomial in q$q$ for Γ=Δn$\Gamma = \Delta _n$ (trivially), Γ=Pn$\Gamma = \operatorname{P}_{{n}}$ (by [21, Thm 7]), and Γ=Kn$\Gamma = \operatorname{K}_n$ (for odd q$q$; by [24, Prop. 2.4]).Let v1,…,vn$v_1,\ldots\,,v_n$ be the distinct vertices of a graph Γ$\Gamma$. Let Y$Y$ consist of algebraically independent variables Yij$Y_{ij}$ (over Z$\mathbf {Z}$) indexed by pairs (i,j)$(i,j)$ with 1⩽i<j⩽n$1\leqslant i &lt; j \leqslant n$ and vi∼vj$v_i \sim v_j$. Let BΓ(Y)$B_{\Gamma} (Y)$ be the antisymmetric n×n$n\times n$ matrix whose (i,j)$(i,j)$ entry for i<j$i &lt; j$ is equal to Yij$Y_{ij}$ if vi∼vj$v_i\sim v_j$ and zero otherwise. (That is, BΓ(Y)$B_{\Gamma} (Y)$ is a generic antisymmetric matrix with support constraints defined by Γ$\Gamma$ as in [29].) Let m$m$ be the number of edges of Γ$\Gamma$. Using an arbitrary ordering, relabel our variables as Y=(Y1,…,Ym)$Y = (Y_1,\ldots\,,Y_m)$. Then [24, Thm B] shows that for odd q$q$, up to a factor given by an explicit power of q$q$ (depending on Γ$\Gamma$ and i$i$), ch(Γ,i;q)$\mathrm{ch}(\Gamma ,i;q)$ coincides with #{y∈Fqm:rkFq(BΓ(y))=2i}$\#\!\lbrace y\in \mathbf {F}_q^m : \operatorname{rk}_{\mathbf {F}_q}(B_{\Gamma} (y)) = 2i\rbrace$. In our proof of Theorem A (see Section 4), the number of conjugacy classes of GΓ(Fq)$\mathbf {G}_{\Gamma} (\mathbf {F}_q)$ of given size is similarly expressed in terms of the number of specialisations of given rank of a matrix of linear forms. In that setting, the latter enumeration can be carried out explicitly using algebraic and graph‐theoretic arguments.It is unclear to the author whether such a line of attack could be used to answer Question 1.10. Work of Belkale and Brosnan [2, Thm 0.5] on rank counts for generic symmetric (rather than antisymmetric) matrices with support constraints leads the author to suspect that the functions of q$q$ considered in Question 1.10 might be rather wild as Γ$\Gamma$ and i$i$ vary.OverviewIn Section 2, we relate the definition of graphical groups from Section 1.1 to that from [29]. Introduced in [29], adjacency modules are modules over polynomial rings whose specialisations are closely related to conjugacy classes of graphical groups. In Section 3, we determine the dimensions of such specialisations over fields. By combining this with work of O'Brien and Voll [24], in Section 4, we prove Theorem A. In Section 5, we show that the polynomials CΓ(X,Y)$\mathcal {C}_{\Gamma }(X,Y)$ are well behaved with respect to joins of graphs. In Section 6, we consider the constant term and leading coefficient of CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ in Y$Y$ and we prove Proposition D. Next, Section 7 is devoted to the degree of fΓ(X)$f_{\Gamma} (X)$. Finally, in Section 8, we relate our findings to the study of zeta functions enumerating conjugacy classes.NotationThe symbol ‘⊂$\subset$’ indicates not necessarily proper inclusion. Group commutators are written [x,y]=x−1y−1xy$[x,y] = x^{-1}y^{-1}xy$. For a ring R$R$ and set A$A$, RA$RA$ denotes the free R$R$‐module with basis (ea)a∈A$(\mathsf {e}_a)_{a\in A}$. For x∈RA$x \in RA$, we write x=∑a∈Axaea$x = \sum _{a\in A} x_a \mathsf {e}_a$. We view d×e$d\times e$ matrices over R$R$ as maps Rd→Re$R^d\rightarrow R^e$ acting by right multiplication. We let •$\bullet$ denote a graph with one vertex.GRAPHICAL GROUPS AND GROUP SCHEMESThroughout this section, let Γ=(V,E)$\Gamma = (V,E)$ be a graph with n$n$ vertices and m$m$ edges. We write V={v1,…,vn}$V = \lbrace v_1,\ldots\,,v_n\rbrace$ and J={(j,k):1⩽j<k⩽n,vj∼vk}$J = \lbrace (j,k) : 1\leqslant j &lt; k \leqslant n, v_j\sim v_k\rbrace$. For a ring R$R$, let RΓ$R\, \Gamma$ denote the free R$R$‐module of rank m+n$m+n$ with basis consisting of e1,…,en$\mathsf {e}_1,\ldots\,,\mathsf {e}_n$ and all ejk$\mathsf {e}_{jk}$ for (j,k)∈J$(j,k)\in J$. The chosen ordering of V$V$ allows us to identify RΓ=RV⊕RE$R\,\Gamma = R V \oplus R E$ (see Section 1.8).Graphical groups over quotients of the integersRecall that the right‐angled Artin group associated with the complement of Γ$\Gamma$ isAΓ:=x1,…,xn∣[xi,xj]=1whenevervi≁vj.\begin{equation*} \mathsf {A}_{\Gamma} := {\left\langle x_1,\ldots\,,x_n \mid [x_i,x_j] = 1 \text{ whenever } v_i\notsim v_j \right\rangle} . \end{equation*}Let γ1(H)⩾γ2(H)⩾⋯$\gamma _1(H) \geqslant \gamma _2(H) \geqslant \cdots$ denote the lower central series of a group H$H$. Recall the definition of GΓ(R)$\mathbf {G}_{\Gamma} (R)$ from Section 1.1.2.1Proposition   (i)(Cf. [29, Rem. 3.8].) GΓ(Z)≈AΓ/γ3(AΓ)$\mathbf {G}_{\Gamma} (\mathbf {Z}) \approx \mathsf {A}_{\Gamma} /\gamma _3(\mathsf {A}_{\Gamma} )$.(ii)GΓ(Z/NZ)≈AΓ/γ3(AΓ)AΓN$\mathbf {G}_{\Gamma} (\mathbf {Z}/N\mathbf {Z}) \approx \mathsf {A}_{\Gamma} /\gamma _3(\mathsf {A}_{\Gamma} )\mathsf {A}_{\Gamma} ^N$ if N⩾1$N\geqslant 1$ is an odd integer.ProofPart (i) follows since xi(r)=xi(1)r$x_i(r) = x_i(1)^r$ and zjk(r)=zjk(1)r$z_{jk}(r) = z_{jk}(1)^r$ in GΓ(Z)$\mathbf {G}_{\Gamma} (\mathbf {Z})$ for r∈Z$r\in \mathbf {Z}$, i∈[n]$i\in [n]$, and (j,k)∈J$(j,k)\in J$. Let G=⟨X⟩$G = \langle X \rangle$ be a nilpotent group with γ3(G)=1$\gamma _3(G) = 1$. As is well known (and easy to see), [ab,c]=[a,c][b,c]$[ab,c] = [a,c][b,c]$ and (ab)N=aNbN[b,a]N2$(ab)^N = a^Nb^N [b,a]^{\binom{N}{2}}$ for a,b,c∈G$a,b,c\in G$; cf. [15, III, Hilfssatz 1.2c) and Hilfssatz 1.3b)]. Let N$N$ be odd so that N∣N2$ {N} \mid {\binom{N}{2}}$, Then, if xN=1$x^N = 1$ for all x∈X$x\in X$, we find that aN=1$a^N = 1$ for all a∈G$a\in G$. Taking X={x1(1),…,xn(1)}$X = \lbrace x_1(1),\ldots\,,x_n(1)\rbrace$ and G=GΓ(Z/NZ)$G = \mathbf {G}_{\Gamma} (\mathbf {Z}/N \mathbf {Z})$, we obtain GΓ(Z/NZ)≈GΓ(Z)/⟨x1(1)N,…,xn(1)N⟩≈GΓ(Z)/GΓ(Z)N≈AΓ/γ3(AΓ)AΓN$\mathbf {G}_{\Gamma} (\mathbf {Z}/N\mathbf {Z}) \approx \mathbf {G}_{\Gamma} (\mathbf {Z})/\langle x_1(1)^N,\ldots\,,x_n(1)^N\rangle \approx \mathbf {G}_{\Gamma} (\mathbf {Z})/\mathbf {G}_{\Gamma} (\mathbf {Z})^N \approx \mathsf {A}_{\Gamma} /\gamma _3(\mathsf {A}_{\Gamma} )\mathsf {A}_{\Gamma} ^N$, which proves (ii).□$\Box$Graphical group schemes following [29]We summarise the construction of the graphical group scheme HΓ$\mathbf {H}_{\Gamma}$ from [29, Section 3.4] (denoted by GΓ$\mathbf {G}_{\Gamma}$ in [29]). For a ring R$R$, the underlying set of the group HΓ(R)$\mathbf {H}_{\Gamma} (R)$ is RΓ$R\, \Gamma$. The group operation ∗$*$ is characterised as follows:(G1)0∈RΓ$0\in R\, \Gamma$ is the identity element of HΓ(R)$\mathbf {H}_{\Gamma} (R)$.(G2)For all r1,…,rn∈R$r_1,\ldots\,,r_n\in R$, we have r1e1∗⋯∗rnen=r1e1+⋯+rnen$r_1 \mathsf {e}_1 * \cdots * r_n\mathsf {e}_n = r_1\mathsf {e}_1 + \cdots + r_n \mathsf {e}_n$.(G3)For 1⩽i⩽j⩽n$1 \leqslant i \leqslant j \leqslant n$ and r,s∈R$r,s\in R$, we havesej∗rei=rei+sej−rseij,ifvi∼vj,rei+sej,otherwise.\begin{equation*} s \mathsf {e}_j * r \mathsf {e}_i = {\begin{cases} r \mathsf {e}_i + s \mathsf {e}_j - rs \mathsf {e}_{ij}, & \text{if } v_i\sim v_j,\\ r\mathsf {e}_i + s\mathsf {e}_j, & \text{otherwise.} \end{cases}} \end{equation*}(G4)For all x∈RΓ$x\in R\,\Gamma$ and z∈RE⊂RΓ$z\in RE\subset R\,\Gamma$, we have x∗z=z∗x=x+z$x*z = z*x = x+z$.Given a ring map R→R′$R \rightarrow R^{\prime }$, the induced map RΓ→R′Γ$R\, \Gamma \rightarrow R^{\prime }\, \Gamma$ is a group homomorphism HΓ(R)→HΓ(R′)$\mathbf {H}_{\Gamma} (R) \rightarrow \mathbf {H}_{\Gamma} (R^{\prime })$. The resulting group functor HΓ$\mathbf {H}_{\Gamma}$ represents the graphical group scheme constructed in [29, Section 3.4].Relating the two constructions of graphical group schemesThe group functors GΓ$\mathbf {G}_{\Gamma}$ (see Section 1.1) and HΓ$\mathbf {H}_{\Gamma}$ (see Section 2.2) are naturally isomorphic:2.2PropositionFor each ring R$R$, the map θR:HΓ(R)→GΓ(R)$\theta _R\colon \mathbf {H}_{\Gamma} (R) \rightarrow \mathbf {G}_{\Gamma} (R)$ given by∑i=1nriei+∑(j,k)∈Jrjkejk↦x1(r1)⋯xn(rn)∏(j,k)∈Jzjk(rjk)(ri,rjk∈R)\begin{align*} \sum _{i=1}^n r_i\mathsf {e}_i + \sum _{(j,k)\in J} r_{jk} \mathsf {e}_{jk} &\mapsto x_1(r_1) \cdots x_n(r_n) \prod _{(j,k)\in J} z_{jk}(r_{jk}) & (r_i,r_{jk}\in R) \end{align*}is a group isomorphism. These maps combine to form a natural isomorphism of group functors HΓ→≈GΓ$\mathbf {H}_{\Gamma} \xrightarrow \approx \mathbf {G}_{\Gamma}$.ProofBy a simple calculation in HΓ(R)$\mathbf {H}_{\Gamma} (R)$, we find that for 1⩽i<j⩽n$1\leqslant i &lt; j \leqslant n$ and ri,rj∈R$r_i,r_j\in R$,[riei,rjej]=rirjeij,ifvi∼vj,0,otherwise.\begin{equation*} [r_i \mathsf {e}_i ,r_j \mathsf {e}_j] = {\begin{cases} r_i r_j \mathsf {e}_{ij}, & \text{if } v_i\sim v_j,\\ 0, & \text{otherwise.} \end{cases}} \end{equation*}We thus obtain a group homomorphism πR:GΓ(R)→HΓ(R)$\pi _R\colon \mathbf {G}_{\Gamma} (R) \rightarrow \mathbf {H}_{\Gamma} (R)$ sending each xi(r)$x_i(r)$ to rei$r\mathsf {e}_i$ and each zjk(r)$z_{jk}(r)$ to rejk$r \mathsf {e}_{jk}$. By construction, πRθR=idHΓ(R)$\pi _R \theta _R = \operatorname{id}_{\mathbf {H}_{\Gamma} (R)}$ and θRπR=idHΓ(R)$\theta _R \pi _R = \operatorname{id}_{\mathbf {H}_{\Gamma} (R)}$.□$\Box$We are therefore justified in referring to both GΓ$\mathbf {G}_{\Gamma}$ and HΓ$\mathbf {H}_{\Gamma}$ as ‘the’ graphical group scheme associated with Γ$\Gamma$. As a consequence of Proposition 2.2, each g∈GΓ(R)$g\in \mathbf {G}_{\Gamma} (R)$ admits a unique representationg=x1(r1)⋯xn(rn)∏(j,k)∈Jzjk(rjk).(ri,rjk∈R).\begin{align*} g & = x_1(r_1)\cdots x_n(r_n) \prod _{(j,k)\in J}z_{jk}(r_{jk}). & (r_i,r_{jk}\in R). \end{align*}In particular, GΓ(R)$\mathbf {G}_{\Gamma} (R)$ has order |R|m+n$\vert R\vert ^{m+n}$.Centralisers in graphical groups and graphical Lie algebrasThe graphical Lie algebra hΓ(R)$\mathfrak {h}_{\Gamma} (R)$ associated with Γ$\Gamma$ over a ring R$R$ is defined by endowing the module RΓ$R\, \Gamma$ with the Lie bracket (·,·)$(\,\cdot ,\cdot \,)$ characterised by the following properties:▹$\triangleright$For 1⩽j<k⩽n$1\leqslant j&lt;k\leqslant n$, we have (ej,ek)=ejk$(\mathsf {e}_{j},\mathsf {e}_{k})= \mathsf {e}_{jk}$ if (j,k)∈J$(j,k)\in J$ and (ej,ek)=0$(\mathsf {e}_j,\mathsf {e}_k)= 0$ otherwise.▹$\triangleright$For 1⩽i⩽n$1\leqslant i\leqslant n$ and (j,k),(j′,k′)∈J$(j,k),(j^{\prime },k^{\prime })\in J$, we have (ei,ejk)=(ejk,ej′k′)=0$(\mathsf {e}_i,\mathsf {e}_{jk}) = (\mathsf {e}_{jk},\mathsf {e}_{j^{\prime }k^{\prime }}) = 0$.We may identify hΓ(R)=hΓ(Z)⊗R$\mathfrak {h}_{\Gamma} (R) = \mathfrak {h}_{\Gamma} (\mathbf {Z}) \otimes R$ as Lie R$R$‐algebras and HΓ(R)=hΓ(R)$\mathbf {H}_{\Gamma} (R) = \mathfrak {h}_{\Gamma} (R)$ as sets.Then HΓ$\mathbf {H}_{\Gamma}$ is the group scheme associated with the Lie algebra hΓ(Z)$\mathfrak {h}_{\Gamma} (\mathbf {Z})$ via the construction from [32, Section 2.4.1]; cf. [29, Section 2.4]. In particular, if 2∈R×$2 \in R^\times$, then HΓ(R)$\mathbf {H}_{\Gamma} (R)$ (and hence GΓ(R)$\mathbf {G}_{\Gamma} (R)$) is isomorphic to the group exp(hΓ(R))$\exp (\mathfrak {h}_{\Gamma} (R))$ associated with hΓ(R)$\mathfrak {h}_{\Gamma} (R)$ via the Lazard correspondence. It follows that for an odd prime p$p$, HΓ(Fp)$\mathbf {H}_{\Gamma} (\mathbf {F}_p)$ is isomorphic to the finite p$p$‐group attached to Γ$\Gamma$ by Li and Qiao [17]. He and Qiao [13, Thm 1.1] showed that for graphs Γ$\Gamma$ and Γ′$\Gamma ^{\prime }$ and an odd prime p$p$, HΓ(Fp)$\mathbf {H}_{\Gamma} (\mathbf {F}_p)$ and HΓ′(Fp)$\mathbf {H}_{\Gamma ^{\prime }}(\mathbf {F}_p)$ are isomorphic if and only if Γ$\Gamma$ and Γ′$\Gamma ^{\prime }$ are.2.3Proposition   (i)The group centraliser of h∈RΓ$h \in R\, \Gamma$ in HΓ(R)$\mathbf {H}_{\Gamma} (R)$ and the Lie centraliser of h$h$ in hΓ(R)$\mathfrak {h}_{\Gamma} (R)$ coincide as sets. Hence, the size of each conjugacy class of HΓ(Fq)$\mathbf {H}_{\Gamma} (\mathbf {F}_q)$ is a power of q$q$.(ii)The centres of HΓ(R)$\mathbf {H}_{\Gamma} (R)$ and hΓ(R)$\mathfrak {h}_{\Gamma} (R)$ coincide as sets. The centre of hΓ(R)$\mathfrak {h}_{\Gamma} (R)$ is the submodule of RΓ$R\, \Gamma$ generated by all ejk$\mathsf {e}_{jk}$ for (j,k)∈J$(j,k)\in J$ and all ei$\mathsf {e}_i$ for isolated vertices vi$v_i$.(iii)[HΓ(R),HΓ(R)]=(hΓ(R),hΓ(R))=RE$[\mathbf {H}_{\Gamma} (R),\mathbf {H}_{\Gamma} (R)] = (\mathfrak {h}_{\Gamma} (R),\mathfrak {h}_{\Gamma} (R)) = RE$, and HΓ(R)/[HΓ(R),HΓ(R)]≈RV$\mathbf {H}_{\Gamma} (R)/[\mathbf {H}_{\Gamma} (R),\mathbf {H}_{\Gamma} (R)] \approx R V$.ProofThe elements rei$r \mathsf {e}_i$ for r∈R$r\in R$ and i=1,…,n$i=1,\ldots\,,n$ generate HΓ(R)$\mathbf {H}_{\Gamma} (R)$ as a group and hΓ(R)$\mathfrak {h}_{\Gamma} (R)$ as a Lie R$R$‐algebra. As HΓ(R)$\mathbf {H}_{\Gamma} (R)$ and hΓ(R)$\mathfrak {h}_{\Gamma} (R)$ both have class at most 2, using the calculation from the proof of Proposition 2.2, we find that for h1,h2∈RΓ$h_1,h_2\in R\,\Gamma$, the Lie bracket (h1,h2)$(h_1,h_2)$ coincides with the group commutator [h1,h2]$[h_1,h_2]$. All claims follow easily from this.□$\Box$ADJACENCY MODULESLet Γ=(V,E)$\Gamma = (V,E)$ be a graph. Let XV=(Xv)v∈V$X_V = (X_v)_{v\in V}$ consist of algebraically independent variables over Z$\mathbf {Z}$. The adjacency module of Γ$\Gamma$ is the Z[XV]$\mathbf {Z}[X_V]$‐moduleAdj(Γ):=Z[XV]V⟨Xvew−Xwev:v,w∈Vwithv∼w⟩.\begin{equation*} \operatorname{Adj}(\Gamma ) := \frac{\mathbf {Z}[X_V] V}{\langle X_v \mathsf {e}_w - X_w \mathsf {e}_v : v,w\in V \text{ with } v\sim w\rangle }. \end{equation*}These modules were introduced in [29, Section 3.3]. Their study turns out to be closely related to the enumeration of conjugacy classes of graphical groups; see [29, Sections 3.4, 6, 7]. We note that what we call adjacency modules here are dubbed negative adjacency modules in [29].For a ring R$R$ and x∈RV$x\in R V$, we obtain an R$R$‐module by specialising Adj(Γ)$\operatorname{Adj}(\Gamma )$ in the formAdj(Γ)x:=Adj(Γ)⊗Z[XV]Rx≈RV⟨xvew−xwev:v,w∈Vwithv∼w⟩,\begin{equation*} \operatorname{Adj}(\Gamma )_x := \operatorname{Adj}(\Gamma ) \otimes _{\mathbf {Z}[X_V]} R_x \approx \frac{ R V }{\langle x_v \mathsf {e}_w - x_w \mathsf {e}_v : v,w\in V \text{ with } v\sim w\rangle }, \end{equation*}where Rx$R_x$ denotes R$R$ regarded as a Z[XV]$\mathbf {Z}[X_V]$‐algebra via Xvr=xvr$X_v r = x_v r$ for v∈V$v\in V$ and r∈R$r\in R$.3.1LemmaLet Γi=(Vi,Ei)$\Gamma _i = (V_i,E_i)$ (i=1,2$i=1,2$) be graphs on disjoint vertex sets. Let Γ=Γ1⊕Γ2=(V,E)$\Gamma = \Gamma _1\oplus \Gamma _2 = (V,E)$ be their disjoint union. Let R$R$ be a ring and let x∈RV$x\in R V$. Let xi∈RVi$x_i\in R V_i$ denote the image of x$x$ under the natural projection RV=RV1⊕RV2→RVi$R V = R V_1 \oplus R V_2 \rightarrow R V_i$. Then Adj(Γ)x≈Adj(Γ1)x1⊕Adj(Γ2)x2$\operatorname{Adj}(\Gamma )_x \approx \operatorname{Adj}(\Gamma _1)_{x_1} \oplus \operatorname{Adj}(\Gamma _2)_{x_2}$ as R$R$‐modules.□$\Box$Let K$K$ be a field. For x∈KV$x \in K V$, let supp(x)={v∈V:xv≠0}$\operatorname{supp}(x) = \lbrace v\in V : x_v\not= 0\rbrace$. Recall the definitions of NΓ[U]$\operatorname{N}_{\Gamma} [U]$ and cΓ(U)$\operatorname{c}_{\Gamma} (U)$ from Section 1. The following is a key ingredient of our proof of Theorem A.3.2LemmaLet x∈KV$x\in K V$ and U=supp(x)$U = \operatorname{supp}(x)$. Thendim(Adj(Γ)x)=cΓ(U)+|V|−|NΓ[U]|.\begin{equation*} \dim (\operatorname{Adj}(\Gamma )_x) = {\operatorname{c}_{\Gamma} (U) + \vert V\vert - \vert \operatorname{N}_{\Gamma} [U]\vert }. \end{equation*}ProofLet H:=⟨xvew−xwev:v∼w⟩⩽KV$H := \langle x_v\mathsf {e}_w -x_w\mathsf {e}_v : v\sim w\rangle \leqslant K V$ so that Adj(Γ)x≈KV/H$\operatorname{Adj}(\Gamma )_x\approx K V/H$.(a)Suppose that Γ$\Gamma$ is connected and U=V$U = V$. We need to show that dim(Adj(Γ)x)=1$\dim (\operatorname{Adj}(\Gamma )_x) = 1$. To see that, first note that H⊂x⊥$H \subset x^\perp$, where the orthogonal complement is taken with respect to the bilinear form y·z=∑v∈Vyvzv$y \cdot z = \sum _{v\in V} y_vz_v$. Hence, Adj(Γ)x≠0$\operatorname{Adj}(\Gamma )_x \not= 0$. Choose a spanning tree T$\mathsf {T}$ of Γ$\Gamma$ and a root r∈V$r\in V$. For v∈V∖{r}$v\in V\setminus \lbrace r\rbrace$, let p(v)$\operatorname{p}(v)$ be the predecessor of v$v$ on the unique path from r$r$ to v$v$ in T$\mathsf {T}$. As the elements ev−xvxp(v)ep(v)∈H$\mathsf {e}_v - \frac{x_{v}}{x_{\operatorname{p}(v)} }\mathsf {e}_{\operatorname{p}(v)}\in H$ for v∈V∖{r}$v\in V\setminus \lbrace r\rbrace$ are linearly independent, dim(Adj(Γ)x)⩽1$\dim (\operatorname{Adj}(\Gamma )_x) \leqslant 1$. Thus, dim(Adj(Γ)x)=1$\dim (\operatorname{Adj}(\Gamma )_x) = 1$.(b)If U=V$U = V$ but Γ$\Gamma$ is possibly disconnected, then (a) and Lemma 3.1 show that dim(Adj(Γ)x)=cΓ(U)$\dim (\operatorname{Adj}(\Gamma )_x) = \operatorname{c}_{\Gamma} (U)$ is the number of connected components of Γ$\Gamma$, as claimed.(c)For the general case, let x[U]:=∑u∈Uxueu∈KU$x[U]:= \sum _{u\in U}x_u\mathsf {e}_u \in K U$ be the image of x$x$ under the natural projection KV=KU⊕K(V∖U)→KU$K V = K U \oplus K(V\setminus U) \rightarrow K U$. We claim that Adj(Γ)x≈Adj(Γ[U])x[U]⊕K(V∖NΓ[U])$\operatorname{Adj}(\Gamma )_x \approx \operatorname{Adj}(\Gamma [U])_{x[U]} \oplus K(V\setminus \operatorname{N}_{\Gamma} [U])$. Indeed, this follows since H$H$ is spanned by the following two types of elements:▹$\triangleright$xuev−xveu$x_u \mathsf {e}_v - x_v \mathsf {e}_u$ for adjacent vertices u,v∈U$u,v\in U$.▹$\triangleright$ew$\mathsf {e}_w$ for w∈NΓ[U]∖U$w\in \operatorname{N}_{\Gamma} [U]\setminus U$.The claim follows since by (b), dim(Adj(Γ[U])x[U])=cΓ(U)$\dim (\operatorname{Adj}(\Gamma [U])_{x[U]}) = \operatorname{c}_{\Gamma} (U)$.□$\Box$3.3RemarkLet Γ$\Gamma$ have n$n$ vertices and c$c$ connected components. Lemma 3.2 generalises a well‐known basic fact: each oriented incidence matrix of Γ$\Gamma$ has rank rk(Γ)=n−c$\operatorname{rk}(\Gamma ) = n - c$; see [4, Prop. 4.3]. This easily implies the special case x=∑v∈Vev$x = \sum _{v\in V} \mathsf {e}_v$ of Lemma 3.2.PROOF OF THEOREM AWe first rephrase Theorem A. For a finite group G$G$, define a Dirichlet polynomial ζGcc(s)=∑e=1∞cce(G)e−s$\zeta ^{{\rm cc}}_G(s) = \sum _{e=1}^\infty {\rm cc}_e(G) e^{-s}$; here, s$s$ denotes a complex variable. For almost simple groups, these functions were studied in [18]. Following [19], we refer to ζGcc(s)$\zeta ^{{\rm cc}}_G(s)$ as the conjugacy class zeta function of G$G$. We note that a different notion of conjugacy class zeta functions, occasionally denoted using the same notation ζGcc(s)$\zeta ^{{\rm cc}}_G(s)$, can also be found in the literature; see [3, 7, 27, 28]. Following [29], in Section 8.1, we will refer to the latter functions as class‐counting zeta functions.Let Γ=(V,E)$\Gamma = (V,E)$ be a graph with n$n$ vertices and m$m$ edges. Theorem A is equivalent to ζGΓ(Fq)cc(s)=qmCΓ(q,q−1−s)$\zeta ^{\rm cc}_{\mathbf {G}_{\Gamma} (\mathbf {F}_q)}(s) = q^m \mathcal {C}_{\Gamma} (q,q^{-1-s})$.4.1LemmaζGΓ(Fq)cc(s)=qm−n(s+1)∑x∈FqV|Adj(Γ)x|s+1$ \zeta ^{\rm cc}_{\mathbf {G}_{\Gamma} (\mathbf {F}_q)}(s) = q^{m-n(s+1)}\sum _{x\in \mathbf {F}_q V} \vert \operatorname{Adj}(\Gamma )_x\vert ^{s+1}$.ProofWrite V={v1,…,vn}$V = \lbrace v_1,\ldots\,,v_n\rbrace$ and J={(j,k):1⩽j<k⩽nwithvj∼vk}$J = \lbrace (j,k) : 1 \leqslant j &lt; k\leqslant n \text{ with } v_j \sim v_k\rbrace$; for a ring R$R$, we identify RV=Rn$R V = R^n$. We assume that vn′+1,…,vn$v_{n^{\prime }+1},\ldots\,,v_n$ are the isolated vertices of Γ$\Gamma$. Order the elements of J$J$ lexicographically to establish a bijection between {1,…,m}$\lbrace 1,\ldots\,,m\rbrace$ and J$J$.Write h=hΓ(Z)$\mathfrak {h}= \mathfrak {h}_{\Gamma} (\mathbf {Z})$; see Section 2.4. Let h′$\mathfrak {h}^{\prime }$ and z$\mathfrak {z}$ denote the derived subalgebra and centre of h$\mathfrak {h}$, respectively. By Proposition 2.3, h′$\mathfrak {h}^{\prime }$ and z$\mathfrak {z}$ are free Z$\mathbf {Z}$‐modules of ranks m$m$ and m+n−n′$m + n -n^{\prime }$, respectively. Moreover, the images of e1,…,en′$\mathsf {e}_1,\ldots\,,\mathsf {e}_{n^{\prime }}$ form a Z$\mathbf {Z}$‐basis of h/z$\mathfrak {h}/\mathfrak {z}$. Proposition 2.3 also shows that for each ring R$R$, we may identify h′⊗R$\mathfrak {h}^{\prime } \otimes R$ with the derived subalgebra of h⊗R$\mathfrak {h}\otimes R$, and z⊗R$\mathfrak {z}\otimes R$ with the centre of h⊗R$\mathfrak {h}\otimes R$.Suppose that q=pf$q = p^f$ for an odd prime p$p$. As we noted in Section 2.4, GΓ(Fq)$\mathbf {G}_{\Gamma} (\mathbf {F}_q)$ is isomorphic to the group exp(hΓ(Fq))$\exp (\mathfrak {h}_{\Gamma} (\mathbf {F}_q))$ attached to the Lie Fq$\mathbf {F}_q$‐algebra hΓ(Fq)=h⊗Fq$\mathfrak {h}_{\Gamma} (\mathbf {F}_q) = \mathfrak {h}\otimes \mathbf {F}_q$ via the Lazard correspondence. Let A(X1,…,Xn′)∈Mn′×m(Z[X1,…,Xn′])$A(X_1,\ldots\,,X_{n^{\prime }}) \in \operatorname{M}_{n^{\prime }\times m}(\mathbf {Z}[X_1,\ldots\,,X_{n^{\prime }}])$ be the matrix of linear forms whose (j,k)$(j,k)$th column has precisely two non‐zero entries, namely, Xk$X_k$ and −Xj$-X_j$ in rows j$j$ and k$k$, respectively. Let Zp$\mathbf {Z}_p$ denote the ring of p$p$‐adic integers. It is readily verified that the image of the matrix A(X1,…,Xn′)$A(X_1,\ldots\,,X_{n^{\prime }})$ over Zp[X1,…,Xn′]$\mathbf {Z}_p[X_1,\ldots\,,X_{n^{\prime }}]$ is a ‘commutator matrix’ (as defined in [24, Def. 2.1]) associated with the finite Lie Zp$\mathbf {Z}_p$‐algebra h⊗Fp$\mathfrak {h}\otimes \mathbf {F}_p$.By [24, Thm B]ccqi(GΓ(Fq))=#{x∈Fqn′:rkFq(A(x))=i)}·qn−n′+m−i.\begin{equation*} {\rm cc}_{q^i}(\mathbf {G}_{\Gamma} (\mathbf {F}_q)) = \#\lbrace x\in \mathbf {F}_q^{n^{\prime }} : \operatorname{rk}_{\mathbf {F}_q}(A(x)) = i)\rbrace \cdot q^{n - n^{\prime } + m - i}. \end{equation*}Let Ǎ(X)$\check{A}(X)$ be the m×n$m\times n$ matrix over Z[X]=Z[X1,…,Xn]$\mathbf {Z}[X] = \mathbf {Z}[X_1,\ldots\,,X_n]$ which is obtained from A(X1,…,Xn′)⊤$A(X_1,\ldots\,,X_{n^{\prime }})^\top$ by adding zero columns in positions n′+1,…,n$n^{\prime }+1,\ldots\,,n$. Hence,ccqi(GΓ(Fq))=#{x∈Fqn:rkFq(Ǎ(x))=i)}·qm−i.\begin{equation*} {\rm cc}_{q^i}(\mathbf {G}_{\Gamma} (\mathbf {F}_q)) = \#\lbrace x\in \mathbf {F}_q^{n} : \operatorname{rk}_{\mathbf {F}_q}(\check{A}(x)) = i)\rbrace \cdot q^{m - i}. \end{equation*}By construction, Adj(Γ)x≈Coker(Ǎ(x))$\operatorname{Adj}(\Gamma )_x \approx \operatorname{Coker}(\check{A}(x))$ for all x∈FqV=Fqn$x\in \mathbf {F}_q V = \mathbf {F}_q^n$. In particular, for x∈Fqn$x\in \mathbf {F}_q^n$, we have rkFq(Ǎ(x))=i$\operatorname{rk}_{\mathbf {F}_q}(\check{A}(x)) = i$ if and only if dimFq(Adj(Γ)x)=n−i$\dim _{\mathbf {F}_q}(\operatorname{Adj}(\Gamma )_x) = {n-i}$. Hence, writing αx=|Adj(Γ)x|$\alpha _x = \vert \operatorname{Adj}(\Gamma )_x\vert$, we have rkFq(Ǎ(x))=i$\operatorname{rk}_{\mathbf {F}_q}(\check{A}(x)) = i$ if and only if qnαx−1=qi$q^n\alpha _x^{-1} = q^i$. Thus,ζGΓ(Fq)cc(s)=∑i=0∞ccqi(GΓ(Fq))q−is=∑x∈FqVqm−nαx·(qnαx−1)−s=qm−n(s+1)∑x∈FqVαxs+1.\begin{align*} \zeta ^{{\rm cc}}_{\mathbf {G}_{\Gamma} (\mathbf {F}_q)}(s) & = \sum _{i=0}^\infty {\rm cc}_{q^i}(\mathbf {G}_{\Gamma} (\mathbf {F}_q)) q^{-is} = \sum _{x\in \mathbf {F}_q V} q^{m-n}\alpha _x \cdot (q^n\alpha _x^{-1})^{-s}\\ & = q^{m-n(s+1)} \sum _{x\in \mathbf {F}_qV} \alpha _x^{s+1}. \end{align*}Finally, if q$q$ is even, while the statement of [24, Thm B] itself is no longer directly applicable (due to its reliance on the Lazard correspondence), its proof in [24, Sections 3.1, 3.3–3.4] does apply in the present setting, completing the present proof. Indeed, the key ingredient that we need is to be able to identify GΓ(Fq)$\mathbf {G}_{\Gamma} (\mathbf {F}_q)$ and h⊗Fq$\mathfrak {h}\otimes \mathbf {F}_q$ as sets such that two elements commute in the group if and only if they commute in the Lie algebra. These conditions are satisfied by Propositions 2.2 and 2.3.□$\Box$AProof of TheoremBy combining Lemma 4.1 and Lemma 3.2, we obtainζGΓ(Fq)cc(s)=qm−n(s+1)∑x∈FqVqcΓ(supp(x))+n−|NΓ[supp(x)]|s+1=qm∑x∈FqV(q−1−s)|NΓ[supp(x)]|−cΓ(supp(x))=qm∑U⊂V(q−1)|U|(q−1−s)|NΓ[U]|−cΓ(U)=qmCΓ(q,q−1−s).\begin{align*} \zeta ^{\rm cc}_{\mathbf {G}_{\Gamma} (\mathbf {F}_q)}(s) & = q^{m-n(s+1)} \sum _{x\in \mathbf {F}_q V} {\left(q^{\operatorname{c}_{\Gamma} (\operatorname{supp}(x)) + n -\vert \operatorname{N}_{\Gamma} [\operatorname{supp}(x)]\vert }\right)}^{s+1} \\ & = q^m \sum _{x\in \mathbf {F}_q V} (q^{-1-s})^{\vert \operatorname{N}_{\Gamma} [\operatorname{supp}(x)]\vert - \operatorname{c}_{\Gamma} (\operatorname{supp}(x))} \\ & = q^m \sum _{U\subset V} (q-1)^{\vert U\vert }(q^{-1-s})^{\vert \operatorname{N}_{\Gamma} [U]\vert - \operatorname{c}_{\Gamma} (U)} \\ & = q^m \mathcal {C}_{\Gamma} (q,q^{-1-s}).\\[-42pt] \end{align*}□$\Box$GRAPH OPERATIONS: DISJOINT UNIONS AND JOINSLet Γ1=(V1,E1)$\Gamma _1 = (V_1,E_1)$ and Γ2=(V2,E2)$\Gamma _2 = (V_2,E_2)$ be graphs with V1∩V2=∅$V_1\cap V_2 = \emptyset$. Let Γi$\Gamma _i$ have ni$n_i$ vertices and mi$m_i$ edges. The disjoint union Γ1⊕Γ2$\Gamma _1\oplus \Gamma _2$ and join Γ1∨Γ2$\Gamma _1\vee \Gamma _2$ (see Section 1.2) of Γ1$\Gamma _1$ and Γ2$\Gamma _2$ are both graphs on the vertex set V1∪V2$V_1\cup V_2$ with m1+m2$m_1+m_2$ and m1+m2+n1n2$m_1+ m_2 + n_1n_2$ edges, respectively.5.1PropositionCΓ1⊕Γ2(X,Y)=CΓ1(X,Y)CΓ2(X,Y)$\mathcal {C}_{\Gamma _1\oplus \Gamma _2}(X,Y) = \mathcal {C}_{\Gamma _1}(X,Y) \mathcal {C}_{\Gamma _2}(X,Y)$.ProofThis follows since if Ui⊂Vi$U_i\subset V_i$ for i=1,2$i=1,2$, then NΓ1⊕Γ2[U1∪U2]=NΓ1[U1]∪NΓ2[U2]$\operatorname{N}_{\Gamma _1\oplus \Gamma _2}[U_1\cup U_2] = \operatorname{N}_{\Gamma _1}[U_1] \cup \operatorname{N}_{\Gamma _2}[U_2]$ and cΓ1⊕Γ2(U1∪U2)=cΓ1(U1)+cΓ2(U2)$\operatorname{c}_{\Gamma _1\oplus \Gamma _2}(U_1\cup U_2) = \operatorname{c}_{\Gamma _1}(U_1) + \operatorname{c}_{\Gamma _2}(U_2)$.□$\Box$Proposition 5.1 also follows, a fortiori, from Theorem A and the identity cce(G1×G2)=∑d∣eccd(G1)cce/d(G2)${\rm cc}_{e}(G_1\times G_2) = \sum \limits _{ {d} \mid {e} } {\rm cc}_{d}(G_1){\rm cc}_{e/d}(G_2)$ for finite groups G1$G_1$ and G2$G_2$.5.2PropositionCΓ1∨Γ2(X,Y)=1+CΓ1(X,Y)−1Yn2+Yn1CΓ2(X,Y)−1+(Xn1−1)(Xn2−1)Yn1+n2−1.\begin{equation*} \mathcal {C}_{\Gamma _1\vee \Gamma _2}(X,Y) = 1 + \Bigl (\mathcal {C}_{\Gamma _1}(X,Y)-1\Bigr ) Y^{n_2} + Y^{n_1} \Bigl (\mathcal {C}_{\Gamma _2}(X,Y)-1\Bigr ) + (X^{n_1}-1)(X^{n_2}-1)Y^{n_1+n_2-1}. \end{equation*}ProofWrite Γ=Γ1∨Γ2$\Gamma = \Gamma _1\vee \Gamma _2$ and V=V1∪V2$V = V_1\cup V_2$. Let Ui⊂Vi$U_i \subset V_i$ for i=1,2$i=1,2$ and U=U1∪U2$U = U_1\cup U_2$. We seek to relate the summand t(U):=(X−1)|U|Y|NΓ[U]|−cΓ(U)$t(U) := (X-1)^{\vert U\vert } Y^{\vert \operatorname{N}_{\Gamma} [U]\vert -\operatorname{c}_{\Gamma} (U)}$ in the definition of CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ to the summands ti(Ui):=(X−1)|Ui|Y|NΓi[Ui]|−cΓi(Ui)$t_i(U_i) := (X-1)^{\vert U_i\vert } Y^{\vert \operatorname{N}_{\Gamma _i}[U_i]\vert -\operatorname{c}_{\Gamma _i}(U_i)}$. We consider four cases:1.If U1=U2=U=∅$U_1 = U_2 = U = \emptyset$, then t(U)=1$t(U) = 1$.2.If U1≠∅=U2$U_1 \not= \emptyset = U_2$, then NΓ[U]=NΓ1[U1]∪V2$\operatorname{N}_{\Gamma} [U] = \operatorname{N}_{\Gamma _1}[U_1] \cup V_2$, cΓ(U)=cΓ1(U1)$\operatorname{c}_{\Gamma} (U) = \operatorname{c}_{\Gamma _1}(U_1)$, and t(U)=t1(U1)Yn2$t(U) = t_1(U_1) Y^{n_2}$.3.Analogously, if U1=∅≠U2$U_1 = \emptyset \not= U_2$, then t(U)=Yn1t2(U2)$t(U) = Y^{n_1} t_2(U_2)$.4.If U1≠∅≠U2$U_1\not= \emptyset \not= U_2$, then NΓ[U]=V$\operatorname{N}_{\Gamma }[U] = V$, cΓ(U)=1$\operatorname{c}_{\Gamma} (U) = 1$, and t(U)=(X−1)|U1|+|U2|Yn1+n2−1$t(U) = (X-1)^{\vert U_1\vert + \vert U_2\vert } Y^{n_1+n_2-1}$.We conclude thatCΓ(X,Y)=1+CΓ1(X,Y)−1Yn2+Yn1CΓ2(X,Y)−1+∑∅≠U1⊂V1∅≠U2⊂V2(X−1)|U1|(X−1)|U2|Yn1+n2−1.\begin{align*} \mathcal {C}_{\Gamma} (X,Y) & = 1 + \Bigl (\mathcal {C}_{\Gamma _1}(X,Y)-1\Bigr ) Y^{n_2} + Y^{n_1} \Bigl (\mathcal {C}_{\Gamma _2}(X,Y)-1\Bigr ) \\ &\quad + {{\left(\sum _{\substack{\emptyset \not= U_1\subset V_1\\ \emptyset \not= U_2\subset V_2}} (X-1)^{\vert U_1\vert } (X-1)^{\vert U_2\vert }\right)}} Y^{n_1 + n_2-1}.\\[-42pt] \end{align*}□$\Box$As we will explain in Section 8.2, Proposition 5.2 is closely related to [29, Prop. 8.4].5.3ExampleComplete bipartite graphsLet Ka,b=Δa∨Δb$\operatorname{K}_{a,b} = \Delta _a \vee \Delta _b$ be a complete bipartite graph. Recall from Example 1.7 that CΔn(X,Y)=Xn$\mathcal {C}_{\Delta _n}(X,Y) = X^n$. Therefore, by Proposition 5.2, CKa,b(X,Y)=1+(Xa−1)Yb+Ya(Xb−1)+(Xa−1)(Xb−1)Ya+b−1$\mathcal {C}_{\operatorname{K}_{a,b}}(X,Y) = 1 + (X^a-1)Y^b + Y^a(X^b-1) + (X^a-1)(X^b-1)Y^{a+b-1}$. Hence,FKa,b(X,Y)=X(a−1)(b−1)(Xa−1)(Xb−1)Ya+b−1=+X(a−1)b(Xa−1)Yb+Xa(b−1)(Xb−1)Ya+Xab\begin{align*} \mathsf {F}_{\operatorname{K}_{a,b}}(X,Y) & = X^{(a-1)(b-1)}(X^a-1)(X^b-1)Y^{a+b-1} \\ & \phantom{=} +X^{(a-1)b}(X^a-1)Y^b + X^{a(b-1)}(X^b-1)Y^a +X^{ab} \end{align*}and fKa,b(X)=X(a−1)(b−1)((Xa−1)(Xb−1)+Xa−1(Xa−1)+Xb−1(Xb−1)+Xa+b−1)$f_{\operatorname{K}_{a,b}}(X) = X^{(a-1)(b-1)}((X^a-1)(X^b-1) + X^{a-1}(X^a-1) + X^{b-1}(X^b-1)+X^{a+b-1})$. We note that the graphical group GKa,b(Z/NZ)$\mathbf {G}_{\operatorname{K}_{a,b}}(\mathbf {Z}/N\mathbf {Z})$ is the maximal quotient of class at most 2 of the free product (Z/NZ)a∗(Z/NZ)b$(\mathbf {Z}/N\mathbf {Z})^a * (\mathbf {Z}/N\mathbf {Z})^b$; see [29, Section 3.4].5.4ExampleStarsAs a special case of Example 5.3, let Starn=Δn∨•=Kn,1$\operatorname{Star}_n = \Delta _{n} \vee \, \bullet = \operatorname{K}_{n,1}$ be a star graph on n+1$n+1$ vertices. Then CStarn(X,Y)=(Xn+1−Xn)Yn+(Xn−1)Y+1$\mathcal {C}_{\operatorname{Star}_n}(X,Y) = (X^{n+1}-X^n)Y^n + (X^n-1)Y + 1$. Hence, FStarn(X,Y)=Xn−1·(X2−X)Yn+(Xn−1)Y+X$\mathsf {F}_{\operatorname{Star}_n}(X,Y) = X^{n-1} \cdot \bigl ((X^2-X) Y^n + (X^n-1)Y + X\bigr )$ and fStarn(X)=Xn−1(Xn+X2−1)$f_{\operatorname{Star}_n}(X) =X^{n-1}(X^n+X^2-1)$.We record the following consequence of Proposition 5.2 for later use.5.5CorollaryLet Γ1$\Gamma _1$ and Γ2$\Gamma _2$ be graphs. Let Γi$\Gamma _i$ have mi$m_i$ edges and ni$n_i$ vertices. Then5.1fΓ1∨Γ2(X)=Xm1+m2+n1n2+Xm2+(n1−1)n2(fΓ1(X)−Xm1)+Xm1+n1(n2−1)(fΓ2(X)−Xm2)+Xm1+m2+(n1−1)(n2−1)(Xn1−1)(Xn2−1).\begin{align} f_{\Gamma _1\vee \Gamma _2}(X) &= X^{m_1 + m_2 + n_1 n_2} + X^{m_2 + (n_1-1)n_2} (f_{\Gamma _1}(X)-X^{m_1}) \nonumber \\ &\quad + X^{m_1 + n_1(n_2-1)} (f_{\Gamma _2}(X)-X^{m_2})\nonumber \\ &\quad + X^{m_1 + m_2 + (n_1-1)(n_2-1)}(X^{n_1}-1)(X^{n_2}-1). \end{align}Beyond disjoint unions and joins, it would be natural to study the effects of other graph operations on the polynomials CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$.THE CONSTANT AND LEADING TERM OF CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$Let Γ=(V,E)$\Gamma = (V,E)$ be a graph with n$n$ vertices and m$m$ edges. In this section, we primarily view CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ as a polynomial in Y$Y$ over Z[X]$\mathbf {Z}[X]$. Its constant term is easily determined.6.1PropositionCΓ(X,0)=Xi$\mathcal {C}_{\Gamma} (X,0) = X^i$, where i$i$ is the number of isolated vertices of Γ$\Gamma$.ProofAs C•(X,Y)=X$\mathcal {C}_\bullet (X,Y) = X$, by Proposition 5.1, CΓ⊕•(X,Y)=X·CΓ(X,Y)$\mathcal {C}_{\Gamma \oplus \bullet }(X,Y) = X\cdot \mathcal {C}_{\Gamma} (X,Y)$. We may thus assume that i=0$i = 0$. Let U⊂V$U\subset V$. As cΓ(U)⩽|U|⩽|NΓ[U]|$\operatorname{c}_{\Gamma} (U) \leqslant \vert U\vert \leqslant \vert \operatorname{N}_{\Gamma} [U]\vert$, we see that |NΓ[U]|=cΓ(U)$\vert \operatorname{N}_{\Gamma} [U]\vert = \operatorname{c}_{\Gamma} (U)$ if and only if U$U$ consists of isolated vertices. This only happens for U=∅$U = \emptyset$ whence CΓ(X,0)=1$\mathcal {C}_{\Gamma} (X,0) = 1$.□$\Box$For a group‐theoretic interpretation of Proposition 6.1, note that qi$q^i$ is the order of the quotient Z(GΓ(Fq))/[GΓ(Fq),GΓ(Fq)]$\operatorname{Z}(\mathbf {G}_{\Gamma} (\mathbf {F}_q))/[\mathbf {G}_{\Gamma} (\mathbf {F}_q),\mathbf {G}_{\Gamma} (\mathbf {F}_q)]$.Recall that rk(Γ)=n−c$\operatorname{rk}(\Gamma ) = n -c$, where c$c$ is the number of connected components of Γ$\Gamma$.6.2PropositiondegY(CΓ(X,Y))=rk(Γ)$\deg _Y(\mathcal {C}_{\Gamma} (X,Y)) = \operatorname{rk}(\Gamma )$.ProofIf Γ′$\Gamma ^{\prime }$ is any subgraph of Γ$\Gamma$, then rk(Γ′)⩽rk(Γ)$\operatorname{rk}(\Gamma ^{\prime }) \leqslant \operatorname{rk}(\Gamma )$. Let U⊂V$U\subset V$ and write U¯=NΓ[U]$\bar{U} = \operatorname{N}_{\Gamma} [U]$. Since every vertex in U¯∖U$\bar{U} \setminus U$ is adjacent to some vertex in U$U$, we have cΓ(U¯)⩽cΓ(U)$\operatorname{c}_{\Gamma} (\bar{U}) \leqslant \operatorname{c}_{\Gamma} (U)$. Hence, |U¯|−cΓ(U)⩽|U¯|−cΓ(U¯)=rk(Γ[U¯])⩽rk(Γ)$\vert \bar{U}\vert - \operatorname{c}_{\Gamma} (U) \leqslant \vert \bar{U}\vert - \operatorname{c}_{\Gamma} (\bar{U}) = \operatorname{rk}(\Gamma [\bar{U}]) \leqslant \operatorname{rk}(\Gamma )$. Thus, degY(CΓ(X,Y))⩽rk(Γ)$\deg _Y(\mathcal {C}_{\Gamma} (X,Y)) \leqslant \operatorname{rk}(\Gamma )$. The summand corresponding to U=V$U = V$ in (1.2) contributes a term XnYrk(Γ)$X^n Y^{\operatorname{rk}(\Gamma )}$ to CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$, and this term cannot be cancelled by a summand arising from any proper subset.□$\Box$For h(X,Y)=∑ijaijXiYj∈Z[X,Y]$h(X,Y) = \sum _{ij} a_{ij} X^iY^j\in \mathbf {Z}[X,Y]$ with aij∈Z$a_{ij}\in \mathbf {Z}$, write h(X,Y)Yj=∑iaijXi$h(X,Y)\Bigl [Y^j\Bigr ] = \sum _{i} a_{ij}X^i$ for the coefficient of Yj$Y^j$ in h(X,Y)$h(X,Y)$, regarded as a polynomial in Y$Y$. We now consider the leading coefficient CΓ(X,Y)Yrk(Γ)$\mathcal {C}_{\Gamma} (X,Y)\Bigl [Y^{\operatorname{rk}(\Gamma )}\Bigr ]$ of CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ as a polynomial in Y$Y$. Recall that a dominating set of Γ$\Gamma$ is a set D⊂V$D\subset V$ with NΓ[D]=V$\operatorname{N}_{\Gamma} [D] = V$. If, in addition, Γ[D]$\Gamma [D]$ is connected, then D$D$ is a connected dominating set. Let Dc(Γ)$\mathfrak {D}^{\mathrm{c}}(\Gamma )$ be the set of connected dominating sets of Γ$\Gamma$. Clearly, Dc(Γ)≠∅$\mathfrak {D}^{\mathrm{c}}(\Gamma ) \not= \emptyset$ if and only if Γ$\Gamma$ is connected.6.3PropositionSuppose that n⩾2$n\geqslant 2$. Then6.1CΓ(X+1,Y)Yn−1=∑D∈Dc(Γ)X|D|.\begin{equation} \mathcal {C}_{\Gamma} (X+1,Y)\Bigl [Y^{n-1} \Bigr ] = \sum \limits _{D \in \mathfrak {D}^{\mathrm{c}}(\Gamma )} X^{\vert D\vert }. \end{equation}ProofLet U⊂V$U\subset V$. As n⩾2$n \geqslant 2$, |NΓ[U]|−cΓ(U)=n−1$\vert \operatorname{N}_{\Gamma} [U]\vert - \operatorname{c}_{\Gamma} (U) = n-1$ if and only if NΓ[U]=V$\operatorname{N}_{\Gamma} [U] = V$ and cΓ(U)=1$\operatorname{c}_{\Gamma} (U) = 1$. The latter two conditions are satisfied if and only if U∈Dc(Γ)$U\in \mathfrak {D}^{\mathrm{c}}(\Gamma )$.□$\Box$6.4RemarkIn [23], the right‐hand side of (6.1) is referred to as the connected domination polynomial of Γ$\Gamma$. These polynomials are relatives of the widely studied domination polynomials of graphs introduced in [1] (where they were called dominating polynomials).6.5CorollarySuppose that Γ$\Gamma$ does not contain isolated vertices. Let V1,…,Vc⊂V$V_1,\ldots\,,V_c\subset V$ be the distinct connected components of Γ$\Gamma$. ThenCΓ(X+1,Y)Yrk(Γ)=∏i=1c∑Di∈Dc(Γ[Vi])X|Di|.\begin{equation*} \mathcal {C}_{\Gamma} (X+1,Y)\Bigl [Y^{\operatorname{rk}(\Gamma )} \Bigr ] = \prod _{i=1}^c \sum \limits _{D_i \in \mathfrak {D}^{\mathrm{c}}(\Gamma [V_i])} X^{\vert D_i\vert }. \end{equation*}The following is well known.6.6Theorem[9, Section A1.1, [GT2]]The problem of deciding, for a given graph Γ$\Gamma$ and k⩾1$k\geqslant 1$, whether Γ$\Gamma$ admits a connected dominating set of cardinality at most k$k$ is NP‐complete.DProof of PropositionCombine Theorem 6.6 and Proposition 6.3.□$\Box$We finish this section by showing that typically FΓ(0,Y)=0$\mathsf {F}_{\Gamma} (0,Y) = 0$. We first record the following consequence of Proposition 6.3:6.7CorollaryLet Γ$\Gamma$ be a tree with n⩾3$n\geqslant 3$ vertices and ℓ$\ell$ leaves. Then CΓ(X,Y)Yn−1=(X−1)n−ℓXℓ$\mathcal {C}_{\Gamma} (X,Y)\Bigl [Y^{n-1}\Bigr ] = (X-1)^{n-\ell } X^\ell$.ProofLet be the set of leaves of Γ$\Gamma$. Using n⩾3$n\geqslant 3$, it is easy to see that . Hence, by Proposition 6.3, .□$\Box$6.8CorollaryLet Γ$\Gamma$ be an arbitrary graph. Then FΓ(0,Y)=0$\mathsf {F}_{\Gamma} (0,Y) = 0$ unless Γ≈K2⊕r$\Gamma \approx \operatorname{K}_2^{\oplus r}$, in which case FΓ(0,Y)=(−1)rYr$\mathsf {F}_{\Gamma} (0,Y) = (-1)^rY^r$.ProofUsing Proposition 5.1 (and its evident analogue for FΓ(X,Y)$\mathsf {F}_{\Gamma} (X,Y)$), we may assume that Γ$\Gamma$ is connected. If m>rk(Γ)$m &gt; \operatorname{rk}(\Gamma )$, then X$X$ divides FΓ(X,Y)$\mathsf {F}_{\Gamma} (X,Y)$ by Proposition 6.2. Thus, suppose that m=rk(Γ)=n−1$m = \operatorname{rk}(\Gamma ) = n-1$, that is, Γ$\Gamma$ is a tree. Since FK1(X,Y)=X$\mathsf {F}_{\operatorname{K}_1}(X,Y) = X$ and FK2(X,Y)=(X2−1)Y+X$\mathsf {F}_{\operatorname{K}_2}(X,Y) = (X^2-1)Y + X$, we may assume that n⩾3$n \geqslant 3$. Corollary 6.7 then implies that FΓ(0,Y)=0$\mathsf {F}_{\Gamma} (0,Y) = 0$.□$\Box$6.9RemarkThe constant term and leading coefficient of CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ as a polynomial in X−1$X-1$ are easily determined: CΓ(1,Y)=1$\mathcal {C}_{\Gamma} (1,Y) = 1$ and CΓ(X+1,Y)=XnYrk(Γ)+O(Xn−1)$\mathcal {C}_{\Gamma} (X+1,Y) = X^n Y^{\operatorname{rk}(\Gamma )} + \mathcal {O}(X^{n-1})$. The constant term of CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ in X$X$, that is, the polynomial CΓ(0,Y)=∑U⊂V(−1)|U|Y|NΓ[U]|−cΓ(U)$\mathcal {C}_{\Gamma} (0,Y) = \sum _{U\subset V}(-1)^{\vert U\vert } Y^{\vert \operatorname{N}_{\Gamma} [U]\vert -\operatorname{c}_{\Gamma} (U)}$, seems to be more mysterious.THE DEGREES OF CLASS‐COUNTING POLYNOMIALSIn this section, we consider the degrees of class‐counting polynomials fΓ(X)=FΓ(X,1)$f_{\Gamma} (X) = \mathsf {F}_{\Gamma} (X,1)$ (see Theorem 1.2 and Corollary B). As before, let Γ=(V,E)$\Gamma = (V,E)$ be a graph with m$m$ edges and n$n$ vertices.Interpreting deg(fΓ(X))$\deg (f_{\Gamma} (X))$: The invariant η(Γ)$\eta (\Gamma )$For U⊂V$U\subset V$, let dΓ(U)=|NΓ[U]∖U|$\operatorname{d}_{\Gamma} (U) = \vert \operatorname{N}_{\Gamma} [U]\setminus U\vert$, the number of vertices in V∖U$V\setminus U$ with a neighbour in U$U$. Recall that cΓ(U)$\operatorname{c}_{\Gamma} (U)$ denotes the number of connected components of Γ[U]$\Gamma [U]$. Define7.1η(Γ)=maxU⊂VcΓ(U)−dΓ(U)⩾0.\begin{equation} \eta (\Gamma ) = \max _{U\subset V}\Bigl (\operatorname{c}_{\Gamma} (U) - \operatorname{d}_{\Gamma} (U)\Bigr ) \geqslant 0. \end{equation}Corollary B implies7.2deg(fΓ(X))=m+η(Γ).\begin{equation} \deg (f_{\Gamma} (X)) = m + \eta (\Gamma ). \end{equation}Our proof of Proposition D does not imply that computing fΓ(X)=XmCΓ(X,X−1)$f_{\Gamma} (X) = X^m \mathcal {C}_{\Gamma} (X,X^{-1})$ is NP‐hard, motivating Question 1.9.7.1QuestionIs there a polynomial‐time algorithm for computing η(Γ)$\eta (\Gamma )$?7.2RemarkThe author is unaware of previous investigations of the numbers η(Γ)$\eta (\Gamma )$ in the literature. At a formal level, η(Γ)$\eta (\Gamma )$ is reminiscent of other graph‐theoretic invariants such as critical independence numbers [37] (which can be computed in polynomial time).In the following, we establish bounds for η(Γ)$\eta (\Gamma )$. Let α(Γ)$\alpha (\Gamma )$ denote the independence number of Γ$\Gamma$, that is, the maximal cardinality of an independent set of vertices. Clearly,7.3η(Γ)⩽maxU⊂VcΓ(U)=α(Γ).\begin{equation} \eta (\Gamma ) \leqslant \max _{U\subset V} \operatorname{c}_{\Gamma} (U) = \alpha (\Gamma ). \end{equation}While η(Γ)$\eta (\Gamma )$ can be much smaller than α(Γ)$\alpha (\Gamma )$ (cf. Proposition 7.5(ii)), the bound η(Γ)⩽α(Γ)$\eta (\Gamma ) \leqslant \alpha (\Gamma )$ will be useful in our proof of Proposition 7.8 below.Let c$c$ be the number of connected components of Γ$\Gamma$. The case U=V$U = V$ in (7.1) shows that η(Γ)⩾c$\eta (\Gamma ) \geqslant c$. Since η(Γ1⊕Γ2)=η(Γ1)+η(Γ2)$\eta (\Gamma _1\oplus \Gamma _2) = \eta (\Gamma _1) + \eta (\Gamma _2)$, we may assume that Γ$\Gamma$ is connected.7.3PropositionLet Γ$\Gamma$ be connected and n⩾4$n\geqslant 4$. Then η(Γ)⩽n−2$\eta (\Gamma ) \leqslant n-2$ with equality if and only if Γ≈Starn−1$\Gamma \approx \operatorname{Star}_{n-1}$.ProofFor U∈{∅,V}$U\in \lbrace \emptyset , V\rbrace$, we have cΓ(U)−dΓ(U)⩽1<n−2$\operatorname{c}_{\Gamma} (U) - \operatorname{d}_{\Gamma} (U) \leqslant 1 &lt; n-2$. Let U⊂V$U\subset V$ with ∅≠U≠V$\emptyset \not= U\not= V$. Then cΓ(U)⩽n−1$\operatorname{c}_{\Gamma} (U) \leqslant n - 1$ and dΓ(U)>0$\operatorname{d}_{\Gamma} (U) &gt; 0$ since Γ$\Gamma$ is connected. Hence, cΓ(U)−dΓ(U)⩽n−2$\operatorname{c}_{\Gamma} (U) - \operatorname{d}_{\Gamma} (U) \leqslant n-2$ and η(Γ)⩽n−2$\eta (\Gamma ) \leqslant n-2$. Moreover, if cΓ(U)−dΓ(U)=n−2$\operatorname{c}_{\Gamma} (U) - \operatorname{d}_{\Gamma} (U) = n-2$, then cΓ(U)=n−1$\operatorname{c}_{\Gamma} (U) = n-1$ and dΓ(U)=1$\operatorname{d}_{\Gamma} (U) = 1$. This is equivalent to Γ$\Gamma$ being a star graph whose centre is the unique vertex in V∖U$V\setminus U$.□$\Box$By Proposition 7.3, η(Γ)$\eta (\Gamma )$ rarely attains its maximal value among graphs with n$n$ vertices. In contrast, η(Γ)=1$\eta (\Gamma ) = 1$ occurs frequently. Note that η(Kn)=1$\eta (\operatorname{K}_n) = 1$ by Example 1.3. For complete bipartite graphs, we obtain the following:7.4Propositionη(Ka,b)=max(1,|a−b|)$\eta (\operatorname{K}_{a,b}) = \max (1,\vert {a-b}\vert )$.ProofThis follows by inspection from the formula for fKa,b(X)$f_{\operatorname{K}_{a,b}}(X)$ in Example 5.3.□$\Box$Hence, η(Ka,b)=1$\eta (\operatorname{K}_{a,b}) = 1$ if and only if |a−b|⩽1$\vert {a-b}\vert \leqslant 1$. To obtain further examples of graphs Γ$\Gamma$ with η(Γ)=1$\eta (\Gamma ) = 1$, recall that a graph is claw‐free if it does not contain K1,3≈Star3$\operatorname{K}_{1,3} \approx \operatorname{Star}_3$ as an induced subgraph. The following proposition and its proof are due to Matteo Cavaleri. The author thanks him for kindly permitting this material to be included here.7.5Proposition   (i)η(Γ)=maxcΓ(U)+|U|−|V|:U⊂VisadominatingsetofΓ$\eta (\Gamma ) = \max \Bigl (\operatorname{c}_{\Gamma} (U) + \vert U\vert - \vert V\vert : U\subset V\text{ is a dominating set of } \Gamma \Bigr )$.(ii)If Γ$\Gamma$ is claw‐free and connected, then η(Γ)=1$\eta (\Gamma ) = 1$ and thus deg(fΓ(X))=m+1$\deg (f_{\Gamma} (X)) = m + 1$.Proof   (i)Let U⊂V$U\subset V$ with NΓ[U]≠V$\operatorname{N}_{\Gamma} [U]\not= V$. Let C⊂V$C\subset V$ be a connected component of Γ[V∖NΓ[U]]$\Gamma [V\setminus \operatorname{N}_{\Gamma} [U]]$. Clearly, cΓ(U∪C)=cΓ(U)+1$\operatorname{c}_{\Gamma} (U\cup C) = \operatorname{c}_{\Gamma} (U) + 1$. Let x∈NΓ[U∪C]∖(U∪C)$x\in \operatorname{N}_{\Gamma} [U\cup C]\setminus (U\cup C)$. Then x∼y$x\sim y$ for some y∈U∪C$y\in U \cup C$. Suppose that x∉NΓ[U]$x\not\in \operatorname{N}_{\Gamma} [U]$ so that y∈C$y\in C$. Then x∈C$x\in C$ by the definition of C$C$. This contradiction shows that NΓ[U∪C]∖(U∪C)⊂NΓ[U]∖U$\operatorname{N}_{\Gamma} [U\cup C]\setminus (U\cup C) \subset \operatorname{N}_{\Gamma} [U]\setminus U$. Hence, cΓ(U∪C)−dΓ(U∪C)>cΓ(U)−dΓ(U)$\operatorname{c}_{\Gamma} (U\cup C) - \operatorname{d}_{\Gamma} (U\cup C) &gt; \operatorname{c}_{\Gamma} (U) - \operatorname{d}_{\Gamma} (U)$. It follows that the maximal value of cΓ(U)−dΓ(U)$\operatorname{c}_{\Gamma} (U)-\operatorname{d}_{\Gamma} (U)$ is attained for a dominating set U$U$; in that case, dΓ(U)=|V|−|U|$\operatorname{d}_{\Gamma} (U) = \vert V\vert - \vert U\vert$.(ii)Let U⊂V$U\subset V$ be a ⊂$\subset$‐maximal dominating set with cΓ(U)+|U|−|V|=η(Γ)$\operatorname{c}_{\Gamma} (U) + \vert U\vert - \vert V\vert = \eta (\Gamma )$. Suppose that U≠V$U\not= V$. Choose x∈V∖U$x\in V\setminus U$. By maximality of U$U$, cΓ(U)+|U|>cΓ(U∪{x})+|U|+1$\operatorname{c}_{\Gamma} (U) + \vert U\vert &gt; \operatorname{c}_{\Gamma} (U\cup \lbrace x\rbrace ) + \vert U\vert + 1$. Hence, there are distinct connected components C1,C2,C3$C_1,C_2,C_3$ of Γ[U]$\Gamma [U]$ such that Γ[C1∪C2∪C3∪{x}]$\Gamma [C_1\cup C_2\cup C_3\cup \lbrace x\rbrace ]$ is connected. Choose ci∈Ci$c_i\in C_i$ with x∼ci$x\sim c_i$. Then Γ[{c1,c2,c3,x}]≈Star3$\Gamma [\lbrace c_1,c_2,c_3,x\rbrace ] \approx \operatorname{Star}_3$. We conclude that if Γ$\Gamma$ is claw‐free and connected, then U=V$U=V$ and thus η(Γ)=1$\eta (\Gamma ) = 1$.□$\Box$Let Δ(Γ)$\Delta (\Gamma )$ denote the maximum vertex degree of Γ$\Gamma$.7.6LemmaLet T$\mathsf {T}$ be a tree. Then η(T)⩾Δ(T)−1$\eta (\mathsf {T}) \geqslant \Delta (\mathsf {T})-1$.ProofLet w1,…,wd$w_1,\ldots\,,w_d$ be the distinct vertices adjacent to a vertex u$u$ of T$\mathsf {T}$. Let Wi$W_i$ consist of wi$w_i$ and all its descendants in the rooted tree (T,u)$(\mathsf {T},u)$. Define W:=W1∪⋯∪Wd$W := W_1\cup \cdots \cup W_d$. By construction, cT(W)=d$\operatorname{c}_\mathsf {T}(W) = d$ and dT(W)=1$\operatorname{d}_\mathsf {T}(W) = 1$ whence η(T)⩾d−1$\eta (\mathsf {T}) \geqslant d - 1$.□$\Box$7.7CorollaryLet T$\mathsf {T}$ be a tree. Then η(T)=1$\eta (\mathsf {T}) = 1$ if and only if T$\mathsf {T}$ is a path.ProofBy Lemma 7.6, η(T)>1$\eta (\mathsf {T}) &gt; 1$ unless T$\mathsf {T}$ is a path. By Proposition 7.5(ii) or Equation (1.1), we have η(Pn)=1$\eta (\operatorname{P}_{{n}}) = 1$.□$\Box$Upper and lower bounds for deg(fΓ(X))$\deg (f_{\Gamma} (X))$We obtain sharp bounds for deg(fΓ(X))$\deg (f_{\Gamma} (X))$ as Γ$\Gamma$ ranges over all graphs with n$n$ vertices.7.8PropositionLet Γ$\Gamma$ be a graph with n⩾1$n \geqslant 1$ vertices. Then n⩽deg(fΓ(X))⩽n2+1$n \leqslant \deg (f_{\Gamma} (X)) \leqslant \binom{n}{2} + 1$. The lower bound is attained if and only if Γ$\Gamma$ is a disjoint union of paths. The upper bound is attained if and only if Γ$\Gamma$ is complete or n=2$n = 2$.Our proof of Proposition 7.8 will rely on an upper bound for independence numbers.7.9Lemma[12]Let Γ$\Gamma$ be a graph with m$m$ edges and n$n$ vertices. Thenα(Γ)⩽12+14+n2−n−2m.\begin{equation*} \alpha (\Gamma ) \leqslant \left\lfloor \frac{1}{2} + \sqrt {\frac{1}{4} + n^2 - n-2m} \right\rfloor . \end{equation*}7.8Proof of PropositionAs before, let m$m$ denote the number of edges of Γ=(V,E)$\Gamma = (V,E)$.(i)Lower bound.We may assume that Γ$\Gamma$ is connected so that m⩾n−1$m\geqslant n-1$. As η(Γ)⩾1$\eta (\Gamma ) \geqslant 1$, Equation (7.2) shows that deg(fΓ(X))⩾n$\deg (f_{\Gamma} (X)) \geqslant n$ with equality if and only if Γ$\Gamma$ is a tree and η(Γ)=1$\eta (\Gamma ) = 1$. By Corollary 7.7, the latter condition is equivalent to Γ≈Pn$\Gamma \approx \operatorname{P}_{{n}}$.(ii)Upper bound.Since deg(fKn(X))=n2+1$\deg (f_{\operatorname{K}_n}(X)) = \binom{n}{2} + 1$ by Example 1.3, it suffices to show that deg(fΓ(X))⩽n2$\deg (f_{\Gamma} (X)) \leqslant \binom{n}{2}$ whenever m<n2$m &lt; \binom{n}{2}$. We may assume that n⩾3$n\geqslant 3$. Writing m=n2−k$m = \binom{n}{2} - k$, Lemma 7.9 shows that α(Γ)⩽⌊12+2k+14⌋$\alpha (\Gamma ) \leqslant \lfloor \frac{1}{2} + \sqrt {2k+\frac{1}{4}} \rfloor$. Hence, if k⩾2$k\geqslant 2$, then α(Γ)⩽k$\alpha (\Gamma ) \leqslant k$. By Equation (7.3), deg(fΓ(X))=m+η(Γ)⩽m+α(Γ)⩽m+k=n2$\deg (f_{\Gamma} (X)) = m + \eta (\Gamma ) \leqslant m + \alpha (\Gamma ) \leqslant m + k = \binom{n}{2}$. For k=1$k = 1$, we have Γ≈Δ2∨Kn−2$\Gamma \approx \Delta _2 \vee \operatorname{K}_{n-2}$ and deg(fΓ(X))=n2$\deg (f_{\Gamma }(X)) = \binom{n}{2}$ by Proposition 7.5(ii). (Alternatively, we may combine Corollary 5.5 and Example 1.3.)□$\Box$APPLICATIONS TO ZETA FUNCTIONS OF GRAPHICAL GROUP SCHEMESWe briefly relate some of our findings to recent work on zeta functions of groups.Reminder: Class‐counting and conjugacy class zeta functionsThe study of zeta functions associated with groups and group‐theoretic counting problems goes back to the influential work of Grunewald et al. [11]. Let G$\mathbf {G}$ be a group scheme of finite type over a compact discrete valuation ring O$\mathfrak {O}$ with maximal ideal P$\mathfrak {P}$. The class‐counting zeta function of G$\mathbf {G}$ is the Dirichlet series ζGk(s)=∑i=0∞k(G(O/Pi))|O/Pi|−s$\zeta ^{\operatorname{k}}_{\mathbf {G}}(s) = \sum _{i=0}^\infty \operatorname{k}(\mathbf {G}(\mathfrak {O}/\mathfrak {P}^i)) {\vert \mathfrak {O}/\mathfrak {P}^i\vert} ^{-s}$. Beginning with work of du Sautoy [7], these and closely related series enumerating conjugacy classes have recently been studied, see [3, 19, 20, 27–29]. Recall the definition of the conjugacy class zeta function ζGcc(s)$\zeta ^{\rm cc}_G(s)$ associated with a finite group G$G$ from Section 4. Lins [19, Def. 1.2] introduced a refinement of ζGk(s)$\zeta ^{\operatorname{k}}_{\mathbf {G}}(s)$, the bivariate conjugacy class zeta function ζGcc(s1,s2)=∑i=0∞ζG(O/Pi)cc(s1)|O/Pi|−s2$\zeta ^{\rm cc}_{\mathbf {G}}(s_1,s_2) =\sum _{i=0}^\infty \zeta _{\mathbf {G}(\mathfrak {O}/\mathfrak {P}^i)}^{\rm cc}(s_1) \vert \mathfrak {O}/\mathfrak {P}^i\vert ^{-s_2}$ of G$\mathbf {G}$ and studied these functions for certain classes of unipotent group schemes; note that ζGk(s)=ζGcc(0,s)$\zeta ^{\operatorname{k}}_{\mathbf {G}}(s) = \zeta ^{\rm cc}_{\mathbf {G}}(0,s)$.Theorem 1.2 is in fact a special case of a far more general result pertaining to class‐counting zeta functions associated with graphical group schemes.8.1TheoremCf. [29, Cor. B]For each graph Γ$\Gamma$, there exists a rational function W∼Γ(X,Y)∈Q(X,Y)${\tilde{W}}_{\Gamma} (X,Y) \in \mathbf {Q}(X,Y)$ with the following property: For each compact discrete valuation ring O$\mathfrak {O}$ with residue field size q$q$, we have ζGΓ⊗Ok(s)=W∼Γ(q,q−s)$\zeta ^{\operatorname{k}}_{\mathbf {G}_{\Gamma} \otimes \mathfrak {O}}(s)= {\tilde{W}}_{\Gamma} (q,q^{-s})$.Theorem 8.1 contains Theorem 1.2 as a special case via W∼Γ(X,Y)=1+fΓ(X)Y+O(Y2)$\tilde{W}_{\Gamma} (X,Y) = 1 + f_{\Gamma} (X) Y + \mathcal {O}(Y^2)$.8.2RemarkIn the present article, we chose to normalise our polynomials and rational functions slightly differently compared to [29]. Namely, what we call W∼Γ(X,Y)$\tilde{W}_{\Gamma} (X,Y)$ here coincides with WΓ−(X,XmY)$W_{\Gamma} ^-(X,X^mY)$ in [29], where m$m$ is the number of edges of Γ$\Gamma$.Class‐counting zeta functions of graphical group schemes and joinsLet Γ1$\Gamma _1$ and Γ2$\Gamma _2$ be graphs with n1$n_1$ and n2$n_2$ vertices and m1$m_1$ and m2$m_2$ edges, respectively. Define a rational function QΓ1,Γ2(X,Y)∈Q(X,Y)$Q_{\Gamma _1,\Gamma _2}(X,Y) \in \mathbf {Q}(X,Y)$ viaQΓ1,Γ2(X,Y)=Xm1+m2+(n1−1)(n2−1)Y−1=+W∼Γ1(X,Xm2+(n1−1)n2Y)·(1−Xm1+m2+(n1−1)n2Y)(1−Xm1+m2+(n1−1)n2+1Y)=+W∼Γ2(X,Xm1+n1(n2−1)Y)·(1−Xm1+m2+n1(n2−1)Y)(1−Xm1+m2+n1(n2−1)+1Y).\begin{align*} Q_{\Gamma _1,\Gamma _2}(X,Y) & = X^{m_1 + m_2 + (n_1-1)(n_2-1)}Y - 1 \\ & \phantom{=} \, + \tilde{W}_{\Gamma _1}(X,X^{m_2 + (n_1-1)n_2}Y)\cdot (1-X^{m_1 + m_2 + (n_1-1)n_2}Y) (1-X^{m_1 + m_2 + (n_1-1)n_2+1}Y)\\ & \phantom{=} \, + \tilde{W}_{\Gamma _2}(X,X^{m_1 + n_1(n_2-1)}Y)\cdot (1-X^{m_1 + m_2 + n_1(n_2-1)}Y) (1-X^{m_1 + m_2 + n_1(n_2-1)+1}Y). \end{align*}Our study of joins in Section 5 was motivated by the following:8.3Theorem[29, Prop. 8.4]Suppose that Γ1$\Gamma _1$ and Γ2$\Gamma _2$ are cographs. Then8.1W∼Γ1∨Γ2(X,Y)=QΓ1,Γ2(X,Y)(1−Xm1+m2+n1n2Y)(1−Xm1+m2+n1n2+1Y).\begin{equation} \tilde{W}_{\Gamma _1\vee \Gamma _2}(X,Y) = \frac{Q_{\Gamma _1,\Gamma _2}(X,Y)}{(1-X^{m_1+m_2+n_1n_2}Y)(1-X^{m_1+m_2+n_1n_2+1}Y)}. \end{equation}It remains unclear whether the assumption that Γ1$\Gamma _1$ and Γ2$\Gamma _2$ be cographs in Theorem 8.3 is truly needed or if it is merely an artefact of the proof given in [29].8.4Question[29, Question 10.1]Does (8.1) hold for arbitrary graphs Γ1$\Gamma _1$ and Γ2$\Gamma _2$?We obtain a positive answer to a (much weaker!) ‘approximate form’ of Question 8.4.8.5PropositionLet Γ1$\Gamma _1$ and Γ2$\Gamma _2$ be arbitrary graphs with n1$n_1$ and n2$n_2$ vertices and m1$m_1$ and m2$m_2$ edges, respectively. Then, regarded as formal power series in Y$Y$ over Q(X)$\mathbf {Q}(X)$, the rational function W∼Γ1∨Γ2(X,Y)$\tilde{W}_{\Gamma _1\vee \Gamma _2}(X,Y)$ and the right‐hand side of (8.1) agree modulo Y2$Y^2$.ProofThis follows from Corollary 5.5: by expanding the right‐hand side of (8.1) as a series in Y$Y$, we find that the coefficient of Y$Y$ is given by the right‐hand side of (5.1).□$\Box$Uniformity and the difficulty of computing zeta functions of groupsFor reasons that are not truly understood at present, many interesting examples of zeta functions associated with group‐theoretic counting problems are ‘(almost) uniform’. As we now recall, the task of symbolically computing such zeta functions is well defined.UniformityBeginning with a global object G$G$ and a type of counting problem, we often obtain (a) associated local objects Gp$G_p$ indexed by primes (or places) p$p$ and (b) associated local zeta functions ζGp(s)$\zeta _{G_p}(s)$. The family (ζGp(s))p$(\zeta _{G_p}(s))_p$ of zeta functions is (almost) uniform if there exists WG(X,Y)∈Q(X,Y)$W_G(X,Y)\in \mathbf {Q}(X,Y)$ such that ζGp(s)=WG(p,p−s)$\zeta _{G_p}(s) = W_G(p,p^{-s})$ for (almost) all p$p$. (Stronger forms of uniformity may also take into account local base extensions or changing the characteristic of compact discrete valuation rings under consideration. Variants apply to multivariate zeta functions such as ζGcc(s1,s2)$\zeta ^{\rm cc}_{\mathbf {G}}(s_1,s_2)$.) It is then natural to seek to devise algorithms for computing WG(X,Y)$W_G(X,Y)$ and to consider the complexity of such algorithms.Numerous computations of (almost) uniform zeta functions associated with groups and related algebraic structures have been recorded in the literature; see, for example, [8]. For a recent example, Carnevale et al. [5] (see also [6]) obtained strong uniformity results for ideal zeta functions of certain nilpotent Lie rings. Their explicit formulae for rational functions as sums over chain complexes involve sums of super‐exponentially many rational functions.The following example illustrates how class‐counting zeta functions associated with graphical group schemes fit the above template for uniformity of zeta functions.8.6ExampleLet G=GΓ$G = \mathbf {G}_{\Gamma}$ be a graphical group scheme. For a prime p$p$, let Zp$\mathbf {Z}_p$ denote the ring of p$p$‐adic integers and let Gp=G⊗Zp$G_p = G \otimes \mathbf {Z}_p$. Writing ζGp(s)=ζGpk(s)$\zeta _{G_p}(s) = \zeta ^{\operatorname{k}}_{G_p}(s)$, the family (ζGp(s))p$(\zeta _{G_p}(s))_p$ is uniform by Theorem 8.1 with WG(X,Y)=W∼Γ(X,Y)$W_G(X,Y) = \tilde{W}_{\Gamma }(X,Y)$. The constructive proof of Theorem 8.1 in [29] gives rise to an algorithm for computing W∼Γ(X,Y)$\tilde{W}_{\Gamma} (X,Y)$ (see [29, Section 9.1]). While no complexity analysis was carried out in [29], this algorithm appears likely to be substantially worse than polynomial‐time. For a cograph Γ$\Gamma$, [29, Thms C–D] combine to produce a formula for W∼Γ(X,Y)$\tilde{W}_{\Gamma} (X,Y)$ as a sum of explicit rational functions, the number of which grows super‐exponentially with the number of vertices of Γ$\Gamma$.Computing bivariate conjugacy class zeta functionsAs indicated (but not spelled out as such) in [29, Section 8.5], Theorem 8.1 admits the following generalisation: given a graph Γ$\Gamma$, there exists W∼Γ(X,Y,Z)∈Q(X,Y,Z)$\tilde{W}_{\Gamma} (X,Y,Z)\in \mathbf {Q}(X,Y,Z)$ such that for all compact discrete valuation rings with residue field size q$q$, ζGΓcc(s1,s2)=W∼Γ(q,q−s1,q−s2)$\zeta ^{\rm cc}_{\mathbf {G}_{\Gamma} }(s_1,s_2) = \tilde{W}_{\Gamma} (q,q^{-s_1},q^{-s_2})$. (Hence, W∼Γ(X,1,Z)=W∼Γ(X,Z)$\tilde{W}_{\Gamma} (X,1,Z) = \tilde{W}_{\Gamma} (X,Z)$.) Suppose that, given Γ$\Gamma$, an oracle provided us with W∼Γ(X,Y,Z)$\tilde{W}_{\Gamma} (X,Y,Z)$ as a reduced fraction of polynomials. Since W∼Γ(X,Y,Z)=1+FΓ(X,Y)Z+O(Z2)$\tilde{W}_{\Gamma} (X,Y,Z) = 1 + \mathsf {F}_{\Gamma} (X,Y) Z + \mathcal {O}(Z^2)$, we may then compute FΓ(X,Y)$\mathsf {F}_{\Gamma} (X,Y)$ by symbolic differentiation. In particular, Proposition D implies that computing W∼Γ(X,Y,Z)$\tilde{W}_{\Gamma} (X,Y,Z)$ is NP‐hard. To the author's knowledge, this is the first non‐trivial lower bound for the difficulty of computing uniform zeta functions associated with groups. We do not presently obtain a similar lower bound for the difficulty of computing W∼Γ(X,Y)$\tilde{W}_{\Gamma} (X,Y)$ since the difficulty of determining fΓ(X)$f_{\Gamma} (X)$ remained unresolved in Section 7.Open problem: Higher congruence levelsIt is an open problem to find a combinatorial formula for the rational functions W∼Γ(X,Y)$\tilde{W}_{\Gamma} (X,Y)$ (or their generalisations W∼Γ(X,Y,Z)$\tilde{W}_{\Gamma} (X,Y,Z)$ from Section 8.3) as Γ$\Gamma$ ranges over all graphs on a given vertex set; cf. [29, Question 1.8(iii)]. Corollary B provides such a formula for the first non‐trivial coefficient of W∼Γ(X,Y)=1+fΓ(X)Y+O(Y2)$\tilde{W}_{\Gamma} (X,Y) = 1 + f_{\Gamma} (X) Y + \mathcal {O}(Y^2)$, and Theorem A provides a formula for the first non‐trivial coefficient of W∼Γ(X,Y,Z)=1+FΓ(X,Y)Z+O(Z2)$\tilde{W}_{\Gamma} (X,Y,Z)= 1 + \mathsf {F}_{\Gamma} (X,Y) Z + \mathcal {O}(Z^2)$. As suggested by one of the anonymous referees, it is natural to ask whether a combinatorial formula of the type considered here can be obtained for the coefficient of Y2$Y^2$ in W∼Γ(X,Y)$\tilde{W}_{\Gamma} (X,Y)$ or of Z2$Z^2$ in W∼Γ(X,Y,Z)$\tilde{W}_{\Gamma} (X,Y,Z)$. These coefficients enumerate conjugacy classes of graphical groups GΓ(O/P2)$\mathbf {G}_{\Gamma} (\mathfrak {O}/\mathfrak {P}^2)$, where O$\mathfrak {O}$ is a compact discrete valuation ring with maximal ideal P$\mathfrak {P}$. The ‘dual’ problem of enumerating characters (see Section 1.6) is related to recent research developments. 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Zhang, Finding critical independent sets and critical vertex subsets are polynomial problems, SIAM J. Discrete Math. 3 (1990), no. 3, 431–438. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Enumerating conjugacy classes of graphical groups over finite fields

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Abstract

INTRODUCTIONGraphical groupsThroughout, graphs are finite, simple, and (unless otherwise indicated) contain at least one vertex. When the reference to an ambient graph is clear, we use ∼$\sim$ to indicate the associated adjacency relation. All rings are associative, commutative, and unital.Let Γ=(V,E)$\Gamma = (V,E)$ be a graph with n$n$ vertices. The graphical group GΓ(R)$\mathbf {G}_{\Gamma} (R)$ associated with Γ$\Gamma$ over a ring R$R$ was defined in [29, Section 3.4]. For a short equivalent description (see Section 2.3), write V={v1,…,vn}$V = \lbrace v_1,\ldots\,,v_n\rbrace$ and let J={(j,k):1⩽j<k⩽n,vj∼vk}$J = \lbrace (j,k) : 1\leqslant j &lt; k \leqslant n,\, v_j\sim v_k\rbrace$. Then GΓ(R)$\mathbf {G}_{\Gamma} (R)$ is generated by symbols x1(r),…,xn(r)$x_1(r),\ldots\,,x_n(r)$ and zjk(r)$z_{jk}(r)$ for (j,k)∈J$(j,k)\in J$ and r∈R$r\in R$, subject to the following defining relations for i,i′∈[n]:={1,…,n}$i,i^{\prime } \in [n] := \lbrace 1,\ldots\,,n\rbrace$, (j,k),(j′,k′)∈J$(j,k), (j^{\prime },k^{\prime })\in J$, and r,r′∈R$r,r^{\prime }\in R$:(i)xi(r)xi(r′)=xi(r+r′)$x_i(r) x_i(r^{\prime }) = x_i(r+r^{\prime })$ and zjk(r)zjk(r′)=zjk(r+r′)$z_{jk}(r)z_{jk}(r^{\prime }) = z_{jk}(r+r^{\prime })$.       (‘scalars')(ii)[xj(r),xk(r′)]=zjk(rr′)$[x_j(r),x_k(r^{\prime })] = z_{jk}(rr^{\prime })$.       (‘adjacent vertices and commutators')(Recall that (j,k)∈J$(j,k) \in J$ so that vj∼vk$v_j \sim v_k$.)(iii)[xi(r),xi′(r′)]=1$[x_i(r),x_{i^{\prime }}(r^{\prime })] = 1$ if vi≁vi′$v_i\notsim v_{i^{\prime }}$.       (‘non‐adjacent vertices and commutators')(iv)[xi(r),zjk(r′)]=[zj′k′(r),zjk(r′)]=1$[x_i(r), z_{jk}(r^{\prime })] = [ z_{j^{\prime }k^{\prime }}(r),z_{jk}(r^{\prime })]= 1$.       (‘centrality of commutators')Note that every ring map R→R′$R\rightarrow R^{\prime }$ induces an evident group homomorphism GΓ(R)→GΓ(R′)$\mathbf {G}_{\Gamma} (R) \rightarrow \mathbf {G}_{\Gamma} (R^{\prime })$. We will see in Section 2.3 that the resulting group functor GΓ$\mathbf {G}_{\Gamma}$ represents the graphical group scheme associated with Γ$\Gamma$ as defined in [29]. The isomorphism type of GΓ$\mathbf {G}_{\Gamma}$ does not depend on the chosen ordering of the vertices of Γ$\Gamma$.1.1ExampleVarious instances and relatives of graphical groups appeared in the literature.(i)GΓ(Z)$\mathbf {G}_{\Gamma} (\mathbf {Z})$ is isomorphic to the maximal nilpotent quotient of class at most 2 of the right‐angled Artin group ⟨x1,…,xn∣[xi,xj]=1whenevervi≁vj⟩$\langle x_1,\ldots\,,x_n \mid [x_i,x_j] = 1 \text{ whenever } v_i\notsim v_j\rangle$; see Section 2.1.(ii)Let Kn$\operatorname{K}_n$ denote a complete graph on n$n$ vertices. Then GKn(Z)$\mathbf {G}_{\operatorname{K}_n}(\mathbf {Z})$ is a free nilpotent group of rank n$n$ and class at most 2. For each odd prime p$p$, the graphical group GKn(Fp)$\mathbf {G}_{\operatorname{K}_n}(\mathbf {F}_p)$ is a free nilpotent group of rank n$n$, exponent dividing p$p$, and class at most 2. (Both statements follow from Proposition 2.1 below. We note that the prime 2 does not play an exceptional role in any of our main results.)(iii)Let Pn$\operatorname{P}_{{n}}$ be a path graph on n$n$ vertices. Let Ud⩽GLd$\operatorname{U}_d\leqslant \operatorname{GL}_d$ be the group scheme of upper unitriangular d×d$d\times d$ matrices. Then for each ring R$R$, the group GPn(R)$\mathbf {G}_{\operatorname{P}_{{n}}}(R)$ is the maximal quotient of class at most 2 of Un+1(R)$\operatorname{U}_{n+1}(R)$; cf. [29, Section 9.4].(iv)Let Δn$\Delta _n$ denote an edgeless graph on n$n$ vertices. Then for each ring R$R$, we may identify GΔn(R)$\mathbf {G}_{\Delta _n}(R)$ and the (abelian) additive group Rn$R^n$.(v)Let p$p$ be an odd prime. Then GΓ(Fp)$\mathbf {G}_{\Gamma} (\mathbf {F}_p)$ is isomorphic to the p$p$‐group attached to the complement of Γ$\Gamma$ via Mekler's construction [22]; cf. Proposition 2.1(ii). Li and Qiao [17] used what they dubbed the Baer‐Lovász–Tutte procedure to attach a finite p$p$‐group to Γ$\Gamma$. Their group is also isomorphic to GΓ(Fp)$\mathbf {G}_{\Gamma} (\mathbf {F}_p)$; see Section 2.4.Known results: Class numbers of graphical groupsLet cce(G)${\rm cc}_e(G)$ denote the number of conjugacy classes of size e$e$ of a finite group G$G$, and let k(G)=∑e=1∞cce(G)$\operatorname{k}(G) = \sum _{e=1}^\infty {\rm cc}_e(G)$ be the class number of G$G$. It is well known that k(GLd(Fq))$\operatorname{k}(\operatorname{GL}_d(\mathbf {F}_q))$ is a polynomial in q$q$ for fixed d$d$; see [31, Chapter 1, Exercise 190]. This article is devoted to the class numbers k(GΓ(Fq))$\operatorname{k}(\mathbf {G}_{\Gamma} (\mathbf {F}_q))$. We first recall known results.1.2Theorem[29, Cor. 1.3]Given a graph Γ$\Gamma$, there exists fΓ(X)∈Q[X]$f_{\Gamma} (X)\in \mathbf {Q}[X]$ such that k(GΓ(Fq))=fΓ(q)$\operatorname{k}(\mathbf {G}_{\Gamma} (\mathbf {F}_q)) = f_{\Gamma} (q)$ for each prime power q$q$.We call fΓ(X)$f_{\Gamma} (X)$ the class‐counting polynomial of Γ$\Gamma$. In [29], Theorem 1.2 is derived from a more general uniformity result [29, Cor. B] for class‐counting zeta functions associated with graphical group schemes; see Section 8.1. Formulae for fΓ(X)$f_{\Gamma} (X)$ when Γ$\Gamma$ has at most 5 vertices can be deduced from the tables in [29, Section 9]. Moreover, several families of class‐counting polynomials have been previously computed in the literature.1.3ExampleO'Brien and Voll [24, Thm 2.6] gave a formula for the number of conjugacy classes of given size of p$p$‐groups derived from free nilpotent Lie algebras via the Lazard correspondence. Using the interpretation of GKn(Fp)$\mathbf {G}_{\operatorname{K}_n}(\mathbf {F}_p)$ in Example 1.1(ii), their formula or, alternatively, work of Ito and Mann [16, Section 1] yields fKn(X)=Xn−12(Xn+Xn−1−1)$f_{\operatorname{K}_n}(X) = X^{\binom{n-1}{2}}(X^n + X^{n-1}-1)$.1.4ExampleIn light of Example 1.1(iii), Marjoram's enumeration [21, Thm 7] of the irreducible characters of given degree of the maximal class‐2 quotients of Ud(Fq)$\operatorname{U}_d(\mathbf {F}_q)$ yields1.1fPn(X)=∑a=0n2n−aaXn−a−1(X−1)a+n−a−1aXn−a−1(X−1)a+1.\begin{equation} f_{\operatorname{P}_{{n}}}(X) = \sum\limits _{a=0}^{\left\lfloor \frac{n}{2}\right\rfloor } {\left(\binom{n-a}{a} X^{n-a-1}(X-1)^a + \binom{n-a-1}{a}X^{n-a-1}(X-1)^{a+1}\right)}. \end{equation}For any graph Γ$\Gamma$, the size of each conjugacy class of GΓ(Fq)$\mathbf {G}_{\Gamma} (\mathbf {F}_q)$ is of the form qi$q^i$; see Proposition 2.3(i). As indicated in [29, Section 8.5], the methods underpinning Theorem 1.2 can be used to strengthen said theorem: each ccqi(GΓ(Fq))${\rm cc}_{q^i}(\mathbf {G}_{\Gamma} (\mathbf {F}_q))$ is a polynomial in q$q$ with rational coefficients. While constructive, the proof of Theorem 1.2 in [29] relies on an elaborate recursion. In particular, no explicit general formulae for the numbers ccqi(GΓ(Fq))${\rm cc}_{q^i}(\mathbf {G}_{\Gamma} (\mathbf {F}_q))$ or the polynomial fΓ(X)$f_{\Gamma} (X)$ have been previously recorded.Recall that the join Γ1∨Γ2$\Gamma _1\vee \Gamma _2$ of graphs Γ1$\Gamma _1$ and Γ2$\Gamma _2$ is obtained from their disjoint union Γ1⊕Γ2$\Gamma _1\oplus \Gamma _2$ by adding edges connecting each vertex of Γ1$\Gamma _1$ to each vertex of Γ2$\Gamma _2$. Further recall that a cograph is any graph that can be obtained from two cographs on fewer vertices by taking disjoint unions or joins, starting with an isolated vertex.1.5TheoremCf. [29, Theorem E]Let Γ$\Gamma$ be a cograph. Then the coefficients of fΓ(X)$f_{\Gamma} (X)$ as a polynomial in X−1$X-1$ are non‐negative integers.Theorems 1.2 and 1.5 are special cases of more general results pertaining to class numbers of graphical groups GΓ(O/Pi)$\mathbf {G}_{\Gamma} (\mathfrak {O}/\mathfrak {P}^i)$, where O$\mathfrak {O}$ is a compact discrete valuation ring with maximal ideal P$\mathfrak {P}$. We will briefly discuss this topic in Section 8.The graph polynomials CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ and FΓ(X,Y)$\mathsf {F}_{\Gamma} (X,Y)$As before, let Γ=(V,E)$\Gamma = (V,E)$ be a graph. Prior to stating our results, we first define what, to the author's knowledge, appears to be a new graph polynomial. For a (not necessarily proper) subset U⊂V$U\subset V$, let Γ[U]$\Gamma [U]$ be the induced subgraph of Γ$\Gamma$ with vertex set U$U$. Let cΓ(U)$\operatorname{c}_{\Gamma} (U)$ denote the number of connected components of Γ[U]$\Gamma [U]$. (We allow U=∅$U = \emptyset$ in which case cΓ(U)=0$\operatorname{c}_{\Gamma} (U) = 0$.) The closed neighbourhood NΓ[v]⊂V$\operatorname{N}_{\Gamma} [v] \subset V$ of v∈V$v\in V$ consists of v$v$ and all vertices adjacent to it. For U⊂V$U\subset V$, write NΓ[U]=⋃u∈UNΓ[u]$\operatorname{N}_{\Gamma} [U] = \bigcup _{u\in U}\operatorname{N}_{\Gamma} [u]$. Define1.2CΓ(X,Y)=∑U⊂V(X−1)|U|Y|NΓ[U]|−cΓ(U)∈Z[X,Y].\begin{equation} \mathcal {C}_{\Gamma} (X,Y) = \sum _{U\subset V} (X-1)^{\vert U\vert }\, Y^{\vert \operatorname{N}_{\Gamma} [U]\vert - \operatorname{c}_{\Gamma} (U)} \in \mathbf {Z}[X,Y]. \end{equation}1.6RemarkWhile CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ resembles the Tutte polynomial of a matroid on the ground set V$V$, it is unclear to the author whether this is more than a formal similarity. Similarly, CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ is reminiscent of the subgraph polynomial [26, Section 3] of Γ$\Gamma$.Let Γ$\Gamma$ have n$n$ vertices, m$m$ edges, and c$c$ connected components. Recall that the (matroid) rank of Γ$\Gamma$ is rk(Γ)=n−c$\operatorname{rk}(\Gamma ) = n-c$. Define the class‐size polynomial of Γ$\Gamma$ to be1.3FΓ(X,Y)=XmCΓ(X,X−1Y)=∑U⊂V(X−1)|U|Xm+cΓ(U)−|NΓ[U]|Y|NΓ[U]|−cΓ(U).\begin{equation} \mathsf {F}_{\Gamma} (X,Y) = X^m \mathcal {C}_{\Gamma} (X,X^{-1}Y) = \sum _{U\subset V} (X-1)^{\vert U\vert } X^{m + \operatorname{c}_{\Gamma} (U) -\vert \operatorname{N}_{\Gamma} [U]\vert } Y^{\vert \operatorname{N}_{\Gamma} [U]\vert -\operatorname{c}_{\Gamma} (U)}. \end{equation}The degree of CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ as a polynomial in Y$Y$ is rk(Γ)$\operatorname{rk}(\Gamma )$; see Proposition 6.2. As rk(Γ)⩽m$\operatorname{rk}(\Gamma ) \leqslant m$, we conclude that FΓ(X,Y)∈Z[X,Y]$\mathsf {F}_{\Gamma} (X,Y) \in \mathbf {Z}[X,Y]$. While CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ and FΓ(X,Y)$\mathsf {F}_{\Gamma} (X,Y)$ determine each other, CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ is often more convenient to work with and FΓ(X,Y)$\mathsf {F}_{\Gamma} (X,Y)$ turns out to be more directly related to the enumeration of conjugacy classes; see Theorem A.1.7Example   (i)CKn(X,Y)=(Xn−1)Yn−1+1$\mathcal {C}_{\operatorname{K}_n}(X,Y) = (X^n -1) Y^{n-1} + 1$ and FKn(X,Y)=Xn2+1−Xn−12Yn−1+Xn2$\mathsf {F}_{\operatorname{K}_n}(X,Y) = \Bigl (X^{\binom{n}{2}+1}-X^{\binom{n-1}{2}}\Bigr )Y^{n-1} + X^{\binom{n}{2}}$.(ii)CΔn(X,Y)=Xn=FΔn(X,Y)$\mathcal {C}_{\Delta _n}(X,Y) = X^n = \mathsf {F}_{\Delta _n}(X,Y)$.Main resultsLet Γ$\Gamma$ be a graph. The main result of this article justifies the term ‘class‐size polynomial’.ATheoremFΓ(q,Y)=∑i=0∞ccqi(GΓ(Fq))Yi$\mathsf {F}_{\Gamma} (q,Y) = \sum _{i=0}^\infty {\rm cc}_{q^i}(\mathbf {G}_{\Gamma} (\mathbf {F}_q)) Y^i$ for each prime power q$q$.Note that ccqi(GΓ(Fq))=0${\rm cc}_{q^i}(\mathbf {G}_{\Gamma} (\mathbf {F}_q)) = 0$ for all sufficiently large i$i$ so Theorem A asserts an equality of polynomials in Y$Y$.1.8ExampleThe formula in [24, Thm 2.6] referred to in Example 1.3 shows that if q$q$ is an odd prime power, then GKn(Fq)$\mathbf {G}_{\operatorname{K}_n}(\mathbf {F}_q)$ has a centre of order qn2$q^{\binom{n}{2}}$ and precisely (qn−1)qn−12$(q^n-1)q^{\binom{n-1}{2}}$ non‐trivial conjugacy classes, all of size qn−1$q^{n-1}$. These numbers agree with Example 1.7(i). Clearly, Example 1.7(ii) agrees with the fact that GΔn(Fq)≈Fqn$\mathbf {G}_{\Delta _n}(\mathbf {F}_q) \approx \mathbf {F}_q^n$ is abelian.Theorem A provides us with the following explicit formula for the class‐counting polynomial fΓ(X)$f_{\Gamma} (X)$ defined in Theorem 1.2:BCorollaryfΓ(X)=FΓ(X,1)=∑U⊂V(X−1)|U|Xm+cΓ(U)−|NΓ[U]|$ f_{\Gamma} (X) = \mathsf {F}_{\Gamma} (X,1) = \sum _{U\subset V} (X-1)^{\vert U\vert }\, X^{m+\operatorname{c}_{\Gamma} (U)-\vert \operatorname{N}_{\Gamma} [U]\vert }$.Note that Corollary B shows that fΓ(X)$f_{\Gamma} (X)$ has integer coefficients. In the spirit of work surrounding Higman's conjecture (see Section 1.5) and Theorem 1.5, Theorem A implies the following refinement of the preceding observation:CCorollaryFor each e⩾1$e\geqslant 1$, the number of conjugacy classes of GΓ(Fq)$\mathbf {G}_{\Gamma} (\mathbf {F}_q)$ of size e$e$ is given by a polynomial in q−1$q-1$ with non‐negative integer coefficients.ProofUse the binomial theorem to expand powers of X=(X−1)+1$X = (X-1) + 1$ in (1.3).□$\Box$It is natural to ask whether the coefficients referred to in Corollary C enumerate meaningful combinatorial objects. Corollary 6.5 will provide a partial answer to this.We shall not endeavour to improve substantially upon the exponential‐time algorithm for computing CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ suggested by Equation (1.2). Indeed, we will obtain the following:DPropositionComputing CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$, and hence also FΓ(X,Y)$\mathsf {F}_{\Gamma} (X,Y)$, is NP‐hard.More precisely, we will see that knowledge of CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ allows us to read off the cardinalities of connected dominating sets of Γ$\Gamma$. The problem of deciding whether a graph admits a connected dominating set of cardinality at most a given number is known to be NP‐complete; see Theorem 6.6.By Theorem A and Proposition D, symbolically enumerating the conjugacy classes of given size of GΓ(Fq)$\mathbf {G}_{\Gamma} (\mathbf {F}_q)$ (as a polynomial in q$q$) is NP‐hard. The problem of measuring the difficulty of symbolically enumerating all conjugacy classes of GΓ(Fq)$\mathbf {G}_{\Gamma} (\mathbf {F}_q)$ remains open.1.9QuestionIs computing fΓ(X)$f_{\Gamma} (X)$ NP‐hard?Related work: Around Higman's conjectureRecall that Ud⩽GLd$\operatorname{U}_d\leqslant \operatorname{GL}_d$ denotes the group scheme of upper unitriangular d×d$d\times d$ matrices. A famous conjecture due to G. Higman [14] predicts that k(Ud(Fq))$\operatorname{k}(\operatorname{U}_d(\mathbf {F}_q))$ is given by a polynomial in q$q$ for fixed d$d$. This has been confirmed for d⩽13$d\leqslant 13$ by Vera‐López and Arregi [35] and for d⩽16$d\leqslant 16$ by Pak and Soffer [25]. The former authors also showed that the sizes of conjugacy classes of Ud(Fq)$\operatorname{U}_d(\mathbf {F}_q)$ are of the form qi$q^i$ (see [36, Section 3]) and that ccqi(Ud(Fq))${\rm cc}_{q^i}(\operatorname{U}_d(\mathbf {F}_q))$ is a polynomial in q−1$q-1$ with non‐negative integer coefficients for i⩽d−3$i \leqslant d-3$ (see [34]). Many authors studied variants of Higman's conjecture for unipotent groups derived from various types of algebraic groups; see, for example, [10].While logically independent of the work described here, Higman's conjecture (and the body of research surrounding it) certainly provided motivation for topics considered and results obtained in this article (for example, Corollary C).Open problems: Enumerating characters of graphical groupsLet Irr(G)$\operatorname{Irr}(G)$ denote the set of (ordinary) irreducible characters of a finite group G$G$. It is well known that k(G)=|Irr(G)|$\operatorname{k}(G) = \vert \operatorname{Irr}(G)\vert$ (see, for example,[15, V, Section 5]), and the enumeration of irreducible characters of a group (according to their degrees) has often been studied as a ‘dual’ of the enumeration of conjugacy classes (according to their sizes); see, for example, [18, 24, 28]. For odd q$q$, [24, Thm B] implies that the degree χ(1)$\chi (1)$ of each irreducible character χ$\chi$ of a graphical group GΓ(Fq)$\mathbf {G}_{\Gamma} (\mathbf {F}_q)$ is of the form qi$q^i$.1.10QuestionLet Γ$\Gamma$ be a graph and let i⩾0$i\geqslant 0$ be an integer. How doesch(Γ,i;q):=#χ∈Irr(GΓ(Fq)):χ(1)=qi\begin{equation*} \mathrm{ch}(\Gamma ,i;q) := \#{\left\lbrace \chi \in \operatorname{Irr}(\mathbf {G}_{\Gamma} (\mathbf {F}_q)) : \chi (1) = q^i\right\rbrace} \end{equation*}depend on the prime power q$q$?It is known that ch(Γ,i;q)$\mathrm{ch}(\Gamma ,i;q)$ is a polynomial in q$q$ for Γ=Δn$\Gamma = \Delta _n$ (trivially), Γ=Pn$\Gamma = \operatorname{P}_{{n}}$ (by [21, Thm 7]), and Γ=Kn$\Gamma = \operatorname{K}_n$ (for odd q$q$; by [24, Prop. 2.4]).Let v1,…,vn$v_1,\ldots\,,v_n$ be the distinct vertices of a graph Γ$\Gamma$. Let Y$Y$ consist of algebraically independent variables Yij$Y_{ij}$ (over Z$\mathbf {Z}$) indexed by pairs (i,j)$(i,j)$ with 1⩽i<j⩽n$1\leqslant i &lt; j \leqslant n$ and vi∼vj$v_i \sim v_j$. Let BΓ(Y)$B_{\Gamma} (Y)$ be the antisymmetric n×n$n\times n$ matrix whose (i,j)$(i,j)$ entry for i<j$i &lt; j$ is equal to Yij$Y_{ij}$ if vi∼vj$v_i\sim v_j$ and zero otherwise. (That is, BΓ(Y)$B_{\Gamma} (Y)$ is a generic antisymmetric matrix with support constraints defined by Γ$\Gamma$ as in [29].) Let m$m$ be the number of edges of Γ$\Gamma$. Using an arbitrary ordering, relabel our variables as Y=(Y1,…,Ym)$Y = (Y_1,\ldots\,,Y_m)$. Then [24, Thm B] shows that for odd q$q$, up to a factor given by an explicit power of q$q$ (depending on Γ$\Gamma$ and i$i$), ch(Γ,i;q)$\mathrm{ch}(\Gamma ,i;q)$ coincides with #{y∈Fqm:rkFq(BΓ(y))=2i}$\#\!\lbrace y\in \mathbf {F}_q^m : \operatorname{rk}_{\mathbf {F}_q}(B_{\Gamma} (y)) = 2i\rbrace$. In our proof of Theorem A (see Section 4), the number of conjugacy classes of GΓ(Fq)$\mathbf {G}_{\Gamma} (\mathbf {F}_q)$ of given size is similarly expressed in terms of the number of specialisations of given rank of a matrix of linear forms. In that setting, the latter enumeration can be carried out explicitly using algebraic and graph‐theoretic arguments.It is unclear to the author whether such a line of attack could be used to answer Question 1.10. Work of Belkale and Brosnan [2, Thm 0.5] on rank counts for generic symmetric (rather than antisymmetric) matrices with support constraints leads the author to suspect that the functions of q$q$ considered in Question 1.10 might be rather wild as Γ$\Gamma$ and i$i$ vary.OverviewIn Section 2, we relate the definition of graphical groups from Section 1.1 to that from [29]. Introduced in [29], adjacency modules are modules over polynomial rings whose specialisations are closely related to conjugacy classes of graphical groups. In Section 3, we determine the dimensions of such specialisations over fields. By combining this with work of O'Brien and Voll [24], in Section 4, we prove Theorem A. In Section 5, we show that the polynomials CΓ(X,Y)$\mathcal {C}_{\Gamma }(X,Y)$ are well behaved with respect to joins of graphs. In Section 6, we consider the constant term and leading coefficient of CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ in Y$Y$ and we prove Proposition D. Next, Section 7 is devoted to the degree of fΓ(X)$f_{\Gamma} (X)$. Finally, in Section 8, we relate our findings to the study of zeta functions enumerating conjugacy classes.NotationThe symbol ‘⊂$\subset$’ indicates not necessarily proper inclusion. Group commutators are written [x,y]=x−1y−1xy$[x,y] = x^{-1}y^{-1}xy$. For a ring R$R$ and set A$A$, RA$RA$ denotes the free R$R$‐module with basis (ea)a∈A$(\mathsf {e}_a)_{a\in A}$. For x∈RA$x \in RA$, we write x=∑a∈Axaea$x = \sum _{a\in A} x_a \mathsf {e}_a$. We view d×e$d\times e$ matrices over R$R$ as maps Rd→Re$R^d\rightarrow R^e$ acting by right multiplication. We let •$\bullet$ denote a graph with one vertex.GRAPHICAL GROUPS AND GROUP SCHEMESThroughout this section, let Γ=(V,E)$\Gamma = (V,E)$ be a graph with n$n$ vertices and m$m$ edges. We write V={v1,…,vn}$V = \lbrace v_1,\ldots\,,v_n\rbrace$ and J={(j,k):1⩽j<k⩽n,vj∼vk}$J = \lbrace (j,k) : 1\leqslant j &lt; k \leqslant n, v_j\sim v_k\rbrace$. For a ring R$R$, let RΓ$R\, \Gamma$ denote the free R$R$‐module of rank m+n$m+n$ with basis consisting of e1,…,en$\mathsf {e}_1,\ldots\,,\mathsf {e}_n$ and all ejk$\mathsf {e}_{jk}$ for (j,k)∈J$(j,k)\in J$. The chosen ordering of V$V$ allows us to identify RΓ=RV⊕RE$R\,\Gamma = R V \oplus R E$ (see Section 1.8).Graphical groups over quotients of the integersRecall that the right‐angled Artin group associated with the complement of Γ$\Gamma$ isAΓ:=x1,…,xn∣[xi,xj]=1whenevervi≁vj.\begin{equation*} \mathsf {A}_{\Gamma} := {\left\langle x_1,\ldots\,,x_n \mid [x_i,x_j] = 1 \text{ whenever } v_i\notsim v_j \right\rangle} . \end{equation*}Let γ1(H)⩾γ2(H)⩾⋯$\gamma _1(H) \geqslant \gamma _2(H) \geqslant \cdots$ denote the lower central series of a group H$H$. Recall the definition of GΓ(R)$\mathbf {G}_{\Gamma} (R)$ from Section 1.1.2.1Proposition   (i)(Cf. [29, Rem. 3.8].) GΓ(Z)≈AΓ/γ3(AΓ)$\mathbf {G}_{\Gamma} (\mathbf {Z}) \approx \mathsf {A}_{\Gamma} /\gamma _3(\mathsf {A}_{\Gamma} )$.(ii)GΓ(Z/NZ)≈AΓ/γ3(AΓ)AΓN$\mathbf {G}_{\Gamma} (\mathbf {Z}/N\mathbf {Z}) \approx \mathsf {A}_{\Gamma} /\gamma _3(\mathsf {A}_{\Gamma} )\mathsf {A}_{\Gamma} ^N$ if N⩾1$N\geqslant 1$ is an odd integer.ProofPart (i) follows since xi(r)=xi(1)r$x_i(r) = x_i(1)^r$ and zjk(r)=zjk(1)r$z_{jk}(r) = z_{jk}(1)^r$ in GΓ(Z)$\mathbf {G}_{\Gamma} (\mathbf {Z})$ for r∈Z$r\in \mathbf {Z}$, i∈[n]$i\in [n]$, and (j,k)∈J$(j,k)\in J$. Let G=⟨X⟩$G = \langle X \rangle$ be a nilpotent group with γ3(G)=1$\gamma _3(G) = 1$. As is well known (and easy to see), [ab,c]=[a,c][b,c]$[ab,c] = [a,c][b,c]$ and (ab)N=aNbN[b,a]N2$(ab)^N = a^Nb^N [b,a]^{\binom{N}{2}}$ for a,b,c∈G$a,b,c\in G$; cf. [15, III, Hilfssatz 1.2c) and Hilfssatz 1.3b)]. Let N$N$ be odd so that N∣N2$ {N} \mid {\binom{N}{2}}$, Then, if xN=1$x^N = 1$ for all x∈X$x\in X$, we find that aN=1$a^N = 1$ for all a∈G$a\in G$. Taking X={x1(1),…,xn(1)}$X = \lbrace x_1(1),\ldots\,,x_n(1)\rbrace$ and G=GΓ(Z/NZ)$G = \mathbf {G}_{\Gamma} (\mathbf {Z}/N \mathbf {Z})$, we obtain GΓ(Z/NZ)≈GΓ(Z)/⟨x1(1)N,…,xn(1)N⟩≈GΓ(Z)/GΓ(Z)N≈AΓ/γ3(AΓ)AΓN$\mathbf {G}_{\Gamma} (\mathbf {Z}/N\mathbf {Z}) \approx \mathbf {G}_{\Gamma} (\mathbf {Z})/\langle x_1(1)^N,\ldots\,,x_n(1)^N\rangle \approx \mathbf {G}_{\Gamma} (\mathbf {Z})/\mathbf {G}_{\Gamma} (\mathbf {Z})^N \approx \mathsf {A}_{\Gamma} /\gamma _3(\mathsf {A}_{\Gamma} )\mathsf {A}_{\Gamma} ^N$, which proves (ii).□$\Box$Graphical group schemes following [29]We summarise the construction of the graphical group scheme HΓ$\mathbf {H}_{\Gamma}$ from [29, Section 3.4] (denoted by GΓ$\mathbf {G}_{\Gamma}$ in [29]). For a ring R$R$, the underlying set of the group HΓ(R)$\mathbf {H}_{\Gamma} (R)$ is RΓ$R\, \Gamma$. The group operation ∗$*$ is characterised as follows:(G1)0∈RΓ$0\in R\, \Gamma$ is the identity element of HΓ(R)$\mathbf {H}_{\Gamma} (R)$.(G2)For all r1,…,rn∈R$r_1,\ldots\,,r_n\in R$, we have r1e1∗⋯∗rnen=r1e1+⋯+rnen$r_1 \mathsf {e}_1 * \cdots * r_n\mathsf {e}_n = r_1\mathsf {e}_1 + \cdots + r_n \mathsf {e}_n$.(G3)For 1⩽i⩽j⩽n$1 \leqslant i \leqslant j \leqslant n$ and r,s∈R$r,s\in R$, we havesej∗rei=rei+sej−rseij,ifvi∼vj,rei+sej,otherwise.\begin{equation*} s \mathsf {e}_j * r \mathsf {e}_i = {\begin{cases} r \mathsf {e}_i + s \mathsf {e}_j - rs \mathsf {e}_{ij}, & \text{if } v_i\sim v_j,\\ r\mathsf {e}_i + s\mathsf {e}_j, & \text{otherwise.} \end{cases}} \end{equation*}(G4)For all x∈RΓ$x\in R\,\Gamma$ and z∈RE⊂RΓ$z\in RE\subset R\,\Gamma$, we have x∗z=z∗x=x+z$x*z = z*x = x+z$.Given a ring map R→R′$R \rightarrow R^{\prime }$, the induced map RΓ→R′Γ$R\, \Gamma \rightarrow R^{\prime }\, \Gamma$ is a group homomorphism HΓ(R)→HΓ(R′)$\mathbf {H}_{\Gamma} (R) \rightarrow \mathbf {H}_{\Gamma} (R^{\prime })$. The resulting group functor HΓ$\mathbf {H}_{\Gamma}$ represents the graphical group scheme constructed in [29, Section 3.4].Relating the two constructions of graphical group schemesThe group functors GΓ$\mathbf {G}_{\Gamma}$ (see Section 1.1) and HΓ$\mathbf {H}_{\Gamma}$ (see Section 2.2) are naturally isomorphic:2.2PropositionFor each ring R$R$, the map θR:HΓ(R)→GΓ(R)$\theta _R\colon \mathbf {H}_{\Gamma} (R) \rightarrow \mathbf {G}_{\Gamma} (R)$ given by∑i=1nriei+∑(j,k)∈Jrjkejk↦x1(r1)⋯xn(rn)∏(j,k)∈Jzjk(rjk)(ri,rjk∈R)\begin{align*} \sum _{i=1}^n r_i\mathsf {e}_i + \sum _{(j,k)\in J} r_{jk} \mathsf {e}_{jk} &\mapsto x_1(r_1) \cdots x_n(r_n) \prod _{(j,k)\in J} z_{jk}(r_{jk}) & (r_i,r_{jk}\in R) \end{align*}is a group isomorphism. These maps combine to form a natural isomorphism of group functors HΓ→≈GΓ$\mathbf {H}_{\Gamma} \xrightarrow \approx \mathbf {G}_{\Gamma}$.ProofBy a simple calculation in HΓ(R)$\mathbf {H}_{\Gamma} (R)$, we find that for 1⩽i<j⩽n$1\leqslant i &lt; j \leqslant n$ and ri,rj∈R$r_i,r_j\in R$,[riei,rjej]=rirjeij,ifvi∼vj,0,otherwise.\begin{equation*} [r_i \mathsf {e}_i ,r_j \mathsf {e}_j] = {\begin{cases} r_i r_j \mathsf {e}_{ij}, & \text{if } v_i\sim v_j,\\ 0, & \text{otherwise.} \end{cases}} \end{equation*}We thus obtain a group homomorphism πR:GΓ(R)→HΓ(R)$\pi _R\colon \mathbf {G}_{\Gamma} (R) \rightarrow \mathbf {H}_{\Gamma} (R)$ sending each xi(r)$x_i(r)$ to rei$r\mathsf {e}_i$ and each zjk(r)$z_{jk}(r)$ to rejk$r \mathsf {e}_{jk}$. By construction, πRθR=idHΓ(R)$\pi _R \theta _R = \operatorname{id}_{\mathbf {H}_{\Gamma} (R)}$ and θRπR=idHΓ(R)$\theta _R \pi _R = \operatorname{id}_{\mathbf {H}_{\Gamma} (R)}$.□$\Box$We are therefore justified in referring to both GΓ$\mathbf {G}_{\Gamma}$ and HΓ$\mathbf {H}_{\Gamma}$ as ‘the’ graphical group scheme associated with Γ$\Gamma$. As a consequence of Proposition 2.2, each g∈GΓ(R)$g\in \mathbf {G}_{\Gamma} (R)$ admits a unique representationg=x1(r1)⋯xn(rn)∏(j,k)∈Jzjk(rjk).(ri,rjk∈R).\begin{align*} g & = x_1(r_1)\cdots x_n(r_n) \prod _{(j,k)\in J}z_{jk}(r_{jk}). & (r_i,r_{jk}\in R). \end{align*}In particular, GΓ(R)$\mathbf {G}_{\Gamma} (R)$ has order |R|m+n$\vert R\vert ^{m+n}$.Centralisers in graphical groups and graphical Lie algebrasThe graphical Lie algebra hΓ(R)$\mathfrak {h}_{\Gamma} (R)$ associated with Γ$\Gamma$ over a ring R$R$ is defined by endowing the module RΓ$R\, \Gamma$ with the Lie bracket (·,·)$(\,\cdot ,\cdot \,)$ characterised by the following properties:▹$\triangleright$For 1⩽j<k⩽n$1\leqslant j&lt;k\leqslant n$, we have (ej,ek)=ejk$(\mathsf {e}_{j},\mathsf {e}_{k})= \mathsf {e}_{jk}$ if (j,k)∈J$(j,k)\in J$ and (ej,ek)=0$(\mathsf {e}_j,\mathsf {e}_k)= 0$ otherwise.▹$\triangleright$For 1⩽i⩽n$1\leqslant i\leqslant n$ and (j,k),(j′,k′)∈J$(j,k),(j^{\prime },k^{\prime })\in J$, we have (ei,ejk)=(ejk,ej′k′)=0$(\mathsf {e}_i,\mathsf {e}_{jk}) = (\mathsf {e}_{jk},\mathsf {e}_{j^{\prime }k^{\prime }}) = 0$.We may identify hΓ(R)=hΓ(Z)⊗R$\mathfrak {h}_{\Gamma} (R) = \mathfrak {h}_{\Gamma} (\mathbf {Z}) \otimes R$ as Lie R$R$‐algebras and HΓ(R)=hΓ(R)$\mathbf {H}_{\Gamma} (R) = \mathfrak {h}_{\Gamma} (R)$ as sets.Then HΓ$\mathbf {H}_{\Gamma}$ is the group scheme associated with the Lie algebra hΓ(Z)$\mathfrak {h}_{\Gamma} (\mathbf {Z})$ via the construction from [32, Section 2.4.1]; cf. [29, Section 2.4]. In particular, if 2∈R×$2 \in R^\times$, then HΓ(R)$\mathbf {H}_{\Gamma} (R)$ (and hence GΓ(R)$\mathbf {G}_{\Gamma} (R)$) is isomorphic to the group exp(hΓ(R))$\exp (\mathfrak {h}_{\Gamma} (R))$ associated with hΓ(R)$\mathfrak {h}_{\Gamma} (R)$ via the Lazard correspondence. It follows that for an odd prime p$p$, HΓ(Fp)$\mathbf {H}_{\Gamma} (\mathbf {F}_p)$ is isomorphic to the finite p$p$‐group attached to Γ$\Gamma$ by Li and Qiao [17]. He and Qiao [13, Thm 1.1] showed that for graphs Γ$\Gamma$ and Γ′$\Gamma ^{\prime }$ and an odd prime p$p$, HΓ(Fp)$\mathbf {H}_{\Gamma} (\mathbf {F}_p)$ and HΓ′(Fp)$\mathbf {H}_{\Gamma ^{\prime }}(\mathbf {F}_p)$ are isomorphic if and only if Γ$\Gamma$ and Γ′$\Gamma ^{\prime }$ are.2.3Proposition   (i)The group centraliser of h∈RΓ$h \in R\, \Gamma$ in HΓ(R)$\mathbf {H}_{\Gamma} (R)$ and the Lie centraliser of h$h$ in hΓ(R)$\mathfrak {h}_{\Gamma} (R)$ coincide as sets. Hence, the size of each conjugacy class of HΓ(Fq)$\mathbf {H}_{\Gamma} (\mathbf {F}_q)$ is a power of q$q$.(ii)The centres of HΓ(R)$\mathbf {H}_{\Gamma} (R)$ and hΓ(R)$\mathfrak {h}_{\Gamma} (R)$ coincide as sets. The centre of hΓ(R)$\mathfrak {h}_{\Gamma} (R)$ is the submodule of RΓ$R\, \Gamma$ generated by all ejk$\mathsf {e}_{jk}$ for (j,k)∈J$(j,k)\in J$ and all ei$\mathsf {e}_i$ for isolated vertices vi$v_i$.(iii)[HΓ(R),HΓ(R)]=(hΓ(R),hΓ(R))=RE$[\mathbf {H}_{\Gamma} (R),\mathbf {H}_{\Gamma} (R)] = (\mathfrak {h}_{\Gamma} (R),\mathfrak {h}_{\Gamma} (R)) = RE$, and HΓ(R)/[HΓ(R),HΓ(R)]≈RV$\mathbf {H}_{\Gamma} (R)/[\mathbf {H}_{\Gamma} (R),\mathbf {H}_{\Gamma} (R)] \approx R V$.ProofThe elements rei$r \mathsf {e}_i$ for r∈R$r\in R$ and i=1,…,n$i=1,\ldots\,,n$ generate HΓ(R)$\mathbf {H}_{\Gamma} (R)$ as a group and hΓ(R)$\mathfrak {h}_{\Gamma} (R)$ as a Lie R$R$‐algebra. As HΓ(R)$\mathbf {H}_{\Gamma} (R)$ and hΓ(R)$\mathfrak {h}_{\Gamma} (R)$ both have class at most 2, using the calculation from the proof of Proposition 2.2, we find that for h1,h2∈RΓ$h_1,h_2\in R\,\Gamma$, the Lie bracket (h1,h2)$(h_1,h_2)$ coincides with the group commutator [h1,h2]$[h_1,h_2]$. All claims follow easily from this.□$\Box$ADJACENCY MODULESLet Γ=(V,E)$\Gamma = (V,E)$ be a graph. Let XV=(Xv)v∈V$X_V = (X_v)_{v\in V}$ consist of algebraically independent variables over Z$\mathbf {Z}$. The adjacency module of Γ$\Gamma$ is the Z[XV]$\mathbf {Z}[X_V]$‐moduleAdj(Γ):=Z[XV]V⟨Xvew−Xwev:v,w∈Vwithv∼w⟩.\begin{equation*} \operatorname{Adj}(\Gamma ) := \frac{\mathbf {Z}[X_V] V}{\langle X_v \mathsf {e}_w - X_w \mathsf {e}_v : v,w\in V \text{ with } v\sim w\rangle }. \end{equation*}These modules were introduced in [29, Section 3.3]. Their study turns out to be closely related to the enumeration of conjugacy classes of graphical groups; see [29, Sections 3.4, 6, 7]. We note that what we call adjacency modules here are dubbed negative adjacency modules in [29].For a ring R$R$ and x∈RV$x\in R V$, we obtain an R$R$‐module by specialising Adj(Γ)$\operatorname{Adj}(\Gamma )$ in the formAdj(Γ)x:=Adj(Γ)⊗Z[XV]Rx≈RV⟨xvew−xwev:v,w∈Vwithv∼w⟩,\begin{equation*} \operatorname{Adj}(\Gamma )_x := \operatorname{Adj}(\Gamma ) \otimes _{\mathbf {Z}[X_V]} R_x \approx \frac{ R V }{\langle x_v \mathsf {e}_w - x_w \mathsf {e}_v : v,w\in V \text{ with } v\sim w\rangle }, \end{equation*}where Rx$R_x$ denotes R$R$ regarded as a Z[XV]$\mathbf {Z}[X_V]$‐algebra via Xvr=xvr$X_v r = x_v r$ for v∈V$v\in V$ and r∈R$r\in R$.3.1LemmaLet Γi=(Vi,Ei)$\Gamma _i = (V_i,E_i)$ (i=1,2$i=1,2$) be graphs on disjoint vertex sets. Let Γ=Γ1⊕Γ2=(V,E)$\Gamma = \Gamma _1\oplus \Gamma _2 = (V,E)$ be their disjoint union. Let R$R$ be a ring and let x∈RV$x\in R V$. Let xi∈RVi$x_i\in R V_i$ denote the image of x$x$ under the natural projection RV=RV1⊕RV2→RVi$R V = R V_1 \oplus R V_2 \rightarrow R V_i$. Then Adj(Γ)x≈Adj(Γ1)x1⊕Adj(Γ2)x2$\operatorname{Adj}(\Gamma )_x \approx \operatorname{Adj}(\Gamma _1)_{x_1} \oplus \operatorname{Adj}(\Gamma _2)_{x_2}$ as R$R$‐modules.□$\Box$Let K$K$ be a field. For x∈KV$x \in K V$, let supp(x)={v∈V:xv≠0}$\operatorname{supp}(x) = \lbrace v\in V : x_v\not= 0\rbrace$. Recall the definitions of NΓ[U]$\operatorname{N}_{\Gamma} [U]$ and cΓ(U)$\operatorname{c}_{\Gamma} (U)$ from Section 1. The following is a key ingredient of our proof of Theorem A.3.2LemmaLet x∈KV$x\in K V$ and U=supp(x)$U = \operatorname{supp}(x)$. Thendim(Adj(Γ)x)=cΓ(U)+|V|−|NΓ[U]|.\begin{equation*} \dim (\operatorname{Adj}(\Gamma )_x) = {\operatorname{c}_{\Gamma} (U) + \vert V\vert - \vert \operatorname{N}_{\Gamma} [U]\vert }. \end{equation*}ProofLet H:=⟨xvew−xwev:v∼w⟩⩽KV$H := \langle x_v\mathsf {e}_w -x_w\mathsf {e}_v : v\sim w\rangle \leqslant K V$ so that Adj(Γ)x≈KV/H$\operatorname{Adj}(\Gamma )_x\approx K V/H$.(a)Suppose that Γ$\Gamma$ is connected and U=V$U = V$. We need to show that dim(Adj(Γ)x)=1$\dim (\operatorname{Adj}(\Gamma )_x) = 1$. To see that, first note that H⊂x⊥$H \subset x^\perp$, where the orthogonal complement is taken with respect to the bilinear form y·z=∑v∈Vyvzv$y \cdot z = \sum _{v\in V} y_vz_v$. Hence, Adj(Γ)x≠0$\operatorname{Adj}(\Gamma )_x \not= 0$. Choose a spanning tree T$\mathsf {T}$ of Γ$\Gamma$ and a root r∈V$r\in V$. For v∈V∖{r}$v\in V\setminus \lbrace r\rbrace$, let p(v)$\operatorname{p}(v)$ be the predecessor of v$v$ on the unique path from r$r$ to v$v$ in T$\mathsf {T}$. As the elements ev−xvxp(v)ep(v)∈H$\mathsf {e}_v - \frac{x_{v}}{x_{\operatorname{p}(v)} }\mathsf {e}_{\operatorname{p}(v)}\in H$ for v∈V∖{r}$v\in V\setminus \lbrace r\rbrace$ are linearly independent, dim(Adj(Γ)x)⩽1$\dim (\operatorname{Adj}(\Gamma )_x) \leqslant 1$. Thus, dim(Adj(Γ)x)=1$\dim (\operatorname{Adj}(\Gamma )_x) = 1$.(b)If U=V$U = V$ but Γ$\Gamma$ is possibly disconnected, then (a) and Lemma 3.1 show that dim(Adj(Γ)x)=cΓ(U)$\dim (\operatorname{Adj}(\Gamma )_x) = \operatorname{c}_{\Gamma} (U)$ is the number of connected components of Γ$\Gamma$, as claimed.(c)For the general case, let x[U]:=∑u∈Uxueu∈KU$x[U]:= \sum _{u\in U}x_u\mathsf {e}_u \in K U$ be the image of x$x$ under the natural projection KV=KU⊕K(V∖U)→KU$K V = K U \oplus K(V\setminus U) \rightarrow K U$. We claim that Adj(Γ)x≈Adj(Γ[U])x[U]⊕K(V∖NΓ[U])$\operatorname{Adj}(\Gamma )_x \approx \operatorname{Adj}(\Gamma [U])_{x[U]} \oplus K(V\setminus \operatorname{N}_{\Gamma} [U])$. Indeed, this follows since H$H$ is spanned by the following two types of elements:▹$\triangleright$xuev−xveu$x_u \mathsf {e}_v - x_v \mathsf {e}_u$ for adjacent vertices u,v∈U$u,v\in U$.▹$\triangleright$ew$\mathsf {e}_w$ for w∈NΓ[U]∖U$w\in \operatorname{N}_{\Gamma} [U]\setminus U$.The claim follows since by (b), dim(Adj(Γ[U])x[U])=cΓ(U)$\dim (\operatorname{Adj}(\Gamma [U])_{x[U]}) = \operatorname{c}_{\Gamma} (U)$.□$\Box$3.3RemarkLet Γ$\Gamma$ have n$n$ vertices and c$c$ connected components. Lemma 3.2 generalises a well‐known basic fact: each oriented incidence matrix of Γ$\Gamma$ has rank rk(Γ)=n−c$\operatorname{rk}(\Gamma ) = n - c$; see [4, Prop. 4.3]. This easily implies the special case x=∑v∈Vev$x = \sum _{v\in V} \mathsf {e}_v$ of Lemma 3.2.PROOF OF THEOREM AWe first rephrase Theorem A. For a finite group G$G$, define a Dirichlet polynomial ζGcc(s)=∑e=1∞cce(G)e−s$\zeta ^{{\rm cc}}_G(s) = \sum _{e=1}^\infty {\rm cc}_e(G) e^{-s}$; here, s$s$ denotes a complex variable. For almost simple groups, these functions were studied in [18]. Following [19], we refer to ζGcc(s)$\zeta ^{{\rm cc}}_G(s)$ as the conjugacy class zeta function of G$G$. We note that a different notion of conjugacy class zeta functions, occasionally denoted using the same notation ζGcc(s)$\zeta ^{{\rm cc}}_G(s)$, can also be found in the literature; see [3, 7, 27, 28]. Following [29], in Section 8.1, we will refer to the latter functions as class‐counting zeta functions.Let Γ=(V,E)$\Gamma = (V,E)$ be a graph with n$n$ vertices and m$m$ edges. Theorem A is equivalent to ζGΓ(Fq)cc(s)=qmCΓ(q,q−1−s)$\zeta ^{\rm cc}_{\mathbf {G}_{\Gamma} (\mathbf {F}_q)}(s) = q^m \mathcal {C}_{\Gamma} (q,q^{-1-s})$.4.1LemmaζGΓ(Fq)cc(s)=qm−n(s+1)∑x∈FqV|Adj(Γ)x|s+1$ \zeta ^{\rm cc}_{\mathbf {G}_{\Gamma} (\mathbf {F}_q)}(s) = q^{m-n(s+1)}\sum _{x\in \mathbf {F}_q V} \vert \operatorname{Adj}(\Gamma )_x\vert ^{s+1}$.ProofWrite V={v1,…,vn}$V = \lbrace v_1,\ldots\,,v_n\rbrace$ and J={(j,k):1⩽j<k⩽nwithvj∼vk}$J = \lbrace (j,k) : 1 \leqslant j &lt; k\leqslant n \text{ with } v_j \sim v_k\rbrace$; for a ring R$R$, we identify RV=Rn$R V = R^n$. We assume that vn′+1,…,vn$v_{n^{\prime }+1},\ldots\,,v_n$ are the isolated vertices of Γ$\Gamma$. Order the elements of J$J$ lexicographically to establish a bijection between {1,…,m}$\lbrace 1,\ldots\,,m\rbrace$ and J$J$.Write h=hΓ(Z)$\mathfrak {h}= \mathfrak {h}_{\Gamma} (\mathbf {Z})$; see Section 2.4. Let h′$\mathfrak {h}^{\prime }$ and z$\mathfrak {z}$ denote the derived subalgebra and centre of h$\mathfrak {h}$, respectively. By Proposition 2.3, h′$\mathfrak {h}^{\prime }$ and z$\mathfrak {z}$ are free Z$\mathbf {Z}$‐modules of ranks m$m$ and m+n−n′$m + n -n^{\prime }$, respectively. Moreover, the images of e1,…,en′$\mathsf {e}_1,\ldots\,,\mathsf {e}_{n^{\prime }}$ form a Z$\mathbf {Z}$‐basis of h/z$\mathfrak {h}/\mathfrak {z}$. Proposition 2.3 also shows that for each ring R$R$, we may identify h′⊗R$\mathfrak {h}^{\prime } \otimes R$ with the derived subalgebra of h⊗R$\mathfrak {h}\otimes R$, and z⊗R$\mathfrak {z}\otimes R$ with the centre of h⊗R$\mathfrak {h}\otimes R$.Suppose that q=pf$q = p^f$ for an odd prime p$p$. As we noted in Section 2.4, GΓ(Fq)$\mathbf {G}_{\Gamma} (\mathbf {F}_q)$ is isomorphic to the group exp(hΓ(Fq))$\exp (\mathfrak {h}_{\Gamma} (\mathbf {F}_q))$ attached to the Lie Fq$\mathbf {F}_q$‐algebra hΓ(Fq)=h⊗Fq$\mathfrak {h}_{\Gamma} (\mathbf {F}_q) = \mathfrak {h}\otimes \mathbf {F}_q$ via the Lazard correspondence. Let A(X1,…,Xn′)∈Mn′×m(Z[X1,…,Xn′])$A(X_1,\ldots\,,X_{n^{\prime }}) \in \operatorname{M}_{n^{\prime }\times m}(\mathbf {Z}[X_1,\ldots\,,X_{n^{\prime }}])$ be the matrix of linear forms whose (j,k)$(j,k)$th column has precisely two non‐zero entries, namely, Xk$X_k$ and −Xj$-X_j$ in rows j$j$ and k$k$, respectively. Let Zp$\mathbf {Z}_p$ denote the ring of p$p$‐adic integers. It is readily verified that the image of the matrix A(X1,…,Xn′)$A(X_1,\ldots\,,X_{n^{\prime }})$ over Zp[X1,…,Xn′]$\mathbf {Z}_p[X_1,\ldots\,,X_{n^{\prime }}]$ is a ‘commutator matrix’ (as defined in [24, Def. 2.1]) associated with the finite Lie Zp$\mathbf {Z}_p$‐algebra h⊗Fp$\mathfrak {h}\otimes \mathbf {F}_p$.By [24, Thm B]ccqi(GΓ(Fq))=#{x∈Fqn′:rkFq(A(x))=i)}·qn−n′+m−i.\begin{equation*} {\rm cc}_{q^i}(\mathbf {G}_{\Gamma} (\mathbf {F}_q)) = \#\lbrace x\in \mathbf {F}_q^{n^{\prime }} : \operatorname{rk}_{\mathbf {F}_q}(A(x)) = i)\rbrace \cdot q^{n - n^{\prime } + m - i}. \end{equation*}Let Ǎ(X)$\check{A}(X)$ be the m×n$m\times n$ matrix over Z[X]=Z[X1,…,Xn]$\mathbf {Z}[X] = \mathbf {Z}[X_1,\ldots\,,X_n]$ which is obtained from A(X1,…,Xn′)⊤$A(X_1,\ldots\,,X_{n^{\prime }})^\top$ by adding zero columns in positions n′+1,…,n$n^{\prime }+1,\ldots\,,n$. Hence,ccqi(GΓ(Fq))=#{x∈Fqn:rkFq(Ǎ(x))=i)}·qm−i.\begin{equation*} {\rm cc}_{q^i}(\mathbf {G}_{\Gamma} (\mathbf {F}_q)) = \#\lbrace x\in \mathbf {F}_q^{n} : \operatorname{rk}_{\mathbf {F}_q}(\check{A}(x)) = i)\rbrace \cdot q^{m - i}. \end{equation*}By construction, Adj(Γ)x≈Coker(Ǎ(x))$\operatorname{Adj}(\Gamma )_x \approx \operatorname{Coker}(\check{A}(x))$ for all x∈FqV=Fqn$x\in \mathbf {F}_q V = \mathbf {F}_q^n$. In particular, for x∈Fqn$x\in \mathbf {F}_q^n$, we have rkFq(Ǎ(x))=i$\operatorname{rk}_{\mathbf {F}_q}(\check{A}(x)) = i$ if and only if dimFq(Adj(Γ)x)=n−i$\dim _{\mathbf {F}_q}(\operatorname{Adj}(\Gamma )_x) = {n-i}$. Hence, writing αx=|Adj(Γ)x|$\alpha _x = \vert \operatorname{Adj}(\Gamma )_x\vert$, we have rkFq(Ǎ(x))=i$\operatorname{rk}_{\mathbf {F}_q}(\check{A}(x)) = i$ if and only if qnαx−1=qi$q^n\alpha _x^{-1} = q^i$. Thus,ζGΓ(Fq)cc(s)=∑i=0∞ccqi(GΓ(Fq))q−is=∑x∈FqVqm−nαx·(qnαx−1)−s=qm−n(s+1)∑x∈FqVαxs+1.\begin{align*} \zeta ^{{\rm cc}}_{\mathbf {G}_{\Gamma} (\mathbf {F}_q)}(s) & = \sum _{i=0}^\infty {\rm cc}_{q^i}(\mathbf {G}_{\Gamma} (\mathbf {F}_q)) q^{-is} = \sum _{x\in \mathbf {F}_q V} q^{m-n}\alpha _x \cdot (q^n\alpha _x^{-1})^{-s}\\ & = q^{m-n(s+1)} \sum _{x\in \mathbf {F}_qV} \alpha _x^{s+1}. \end{align*}Finally, if q$q$ is even, while the statement of [24, Thm B] itself is no longer directly applicable (due to its reliance on the Lazard correspondence), its proof in [24, Sections 3.1, 3.3–3.4] does apply in the present setting, completing the present proof. Indeed, the key ingredient that we need is to be able to identify GΓ(Fq)$\mathbf {G}_{\Gamma} (\mathbf {F}_q)$ and h⊗Fq$\mathfrak {h}\otimes \mathbf {F}_q$ as sets such that two elements commute in the group if and only if they commute in the Lie algebra. These conditions are satisfied by Propositions 2.2 and 2.3.□$\Box$AProof of TheoremBy combining Lemma 4.1 and Lemma 3.2, we obtainζGΓ(Fq)cc(s)=qm−n(s+1)∑x∈FqVqcΓ(supp(x))+n−|NΓ[supp(x)]|s+1=qm∑x∈FqV(q−1−s)|NΓ[supp(x)]|−cΓ(supp(x))=qm∑U⊂V(q−1)|U|(q−1−s)|NΓ[U]|−cΓ(U)=qmCΓ(q,q−1−s).\begin{align*} \zeta ^{\rm cc}_{\mathbf {G}_{\Gamma} (\mathbf {F}_q)}(s) & = q^{m-n(s+1)} \sum _{x\in \mathbf {F}_q V} {\left(q^{\operatorname{c}_{\Gamma} (\operatorname{supp}(x)) + n -\vert \operatorname{N}_{\Gamma} [\operatorname{supp}(x)]\vert }\right)}^{s+1} \\ & = q^m \sum _{x\in \mathbf {F}_q V} (q^{-1-s})^{\vert \operatorname{N}_{\Gamma} [\operatorname{supp}(x)]\vert - \operatorname{c}_{\Gamma} (\operatorname{supp}(x))} \\ & = q^m \sum _{U\subset V} (q-1)^{\vert U\vert }(q^{-1-s})^{\vert \operatorname{N}_{\Gamma} [U]\vert - \operatorname{c}_{\Gamma} (U)} \\ & = q^m \mathcal {C}_{\Gamma} (q,q^{-1-s}).\\[-42pt] \end{align*}□$\Box$GRAPH OPERATIONS: DISJOINT UNIONS AND JOINSLet Γ1=(V1,E1)$\Gamma _1 = (V_1,E_1)$ and Γ2=(V2,E2)$\Gamma _2 = (V_2,E_2)$ be graphs with V1∩V2=∅$V_1\cap V_2 = \emptyset$. Let Γi$\Gamma _i$ have ni$n_i$ vertices and mi$m_i$ edges. The disjoint union Γ1⊕Γ2$\Gamma _1\oplus \Gamma _2$ and join Γ1∨Γ2$\Gamma _1\vee \Gamma _2$ (see Section 1.2) of Γ1$\Gamma _1$ and Γ2$\Gamma _2$ are both graphs on the vertex set V1∪V2$V_1\cup V_2$ with m1+m2$m_1+m_2$ and m1+m2+n1n2$m_1+ m_2 + n_1n_2$ edges, respectively.5.1PropositionCΓ1⊕Γ2(X,Y)=CΓ1(X,Y)CΓ2(X,Y)$\mathcal {C}_{\Gamma _1\oplus \Gamma _2}(X,Y) = \mathcal {C}_{\Gamma _1}(X,Y) \mathcal {C}_{\Gamma _2}(X,Y)$.ProofThis follows since if Ui⊂Vi$U_i\subset V_i$ for i=1,2$i=1,2$, then NΓ1⊕Γ2[U1∪U2]=NΓ1[U1]∪NΓ2[U2]$\operatorname{N}_{\Gamma _1\oplus \Gamma _2}[U_1\cup U_2] = \operatorname{N}_{\Gamma _1}[U_1] \cup \operatorname{N}_{\Gamma _2}[U_2]$ and cΓ1⊕Γ2(U1∪U2)=cΓ1(U1)+cΓ2(U2)$\operatorname{c}_{\Gamma _1\oplus \Gamma _2}(U_1\cup U_2) = \operatorname{c}_{\Gamma _1}(U_1) + \operatorname{c}_{\Gamma _2}(U_2)$.□$\Box$Proposition 5.1 also follows, a fortiori, from Theorem A and the identity cce(G1×G2)=∑d∣eccd(G1)cce/d(G2)${\rm cc}_{e}(G_1\times G_2) = \sum \limits _{ {d} \mid {e} } {\rm cc}_{d}(G_1){\rm cc}_{e/d}(G_2)$ for finite groups G1$G_1$ and G2$G_2$.5.2PropositionCΓ1∨Γ2(X,Y)=1+CΓ1(X,Y)−1Yn2+Yn1CΓ2(X,Y)−1+(Xn1−1)(Xn2−1)Yn1+n2−1.\begin{equation*} \mathcal {C}_{\Gamma _1\vee \Gamma _2}(X,Y) = 1 + \Bigl (\mathcal {C}_{\Gamma _1}(X,Y)-1\Bigr ) Y^{n_2} + Y^{n_1} \Bigl (\mathcal {C}_{\Gamma _2}(X,Y)-1\Bigr ) + (X^{n_1}-1)(X^{n_2}-1)Y^{n_1+n_2-1}. \end{equation*}ProofWrite Γ=Γ1∨Γ2$\Gamma = \Gamma _1\vee \Gamma _2$ and V=V1∪V2$V = V_1\cup V_2$. Let Ui⊂Vi$U_i \subset V_i$ for i=1,2$i=1,2$ and U=U1∪U2$U = U_1\cup U_2$. We seek to relate the summand t(U):=(X−1)|U|Y|NΓ[U]|−cΓ(U)$t(U) := (X-1)^{\vert U\vert } Y^{\vert \operatorname{N}_{\Gamma} [U]\vert -\operatorname{c}_{\Gamma} (U)}$ in the definition of CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ to the summands ti(Ui):=(X−1)|Ui|Y|NΓi[Ui]|−cΓi(Ui)$t_i(U_i) := (X-1)^{\vert U_i\vert } Y^{\vert \operatorname{N}_{\Gamma _i}[U_i]\vert -\operatorname{c}_{\Gamma _i}(U_i)}$. We consider four cases:1.If U1=U2=U=∅$U_1 = U_2 = U = \emptyset$, then t(U)=1$t(U) = 1$.2.If U1≠∅=U2$U_1 \not= \emptyset = U_2$, then NΓ[U]=NΓ1[U1]∪V2$\operatorname{N}_{\Gamma} [U] = \operatorname{N}_{\Gamma _1}[U_1] \cup V_2$, cΓ(U)=cΓ1(U1)$\operatorname{c}_{\Gamma} (U) = \operatorname{c}_{\Gamma _1}(U_1)$, and t(U)=t1(U1)Yn2$t(U) = t_1(U_1) Y^{n_2}$.3.Analogously, if U1=∅≠U2$U_1 = \emptyset \not= U_2$, then t(U)=Yn1t2(U2)$t(U) = Y^{n_1} t_2(U_2)$.4.If U1≠∅≠U2$U_1\not= \emptyset \not= U_2$, then NΓ[U]=V$\operatorname{N}_{\Gamma }[U] = V$, cΓ(U)=1$\operatorname{c}_{\Gamma} (U) = 1$, and t(U)=(X−1)|U1|+|U2|Yn1+n2−1$t(U) = (X-1)^{\vert U_1\vert + \vert U_2\vert } Y^{n_1+n_2-1}$.We conclude thatCΓ(X,Y)=1+CΓ1(X,Y)−1Yn2+Yn1CΓ2(X,Y)−1+∑∅≠U1⊂V1∅≠U2⊂V2(X−1)|U1|(X−1)|U2|Yn1+n2−1.\begin{align*} \mathcal {C}_{\Gamma} (X,Y) & = 1 + \Bigl (\mathcal {C}_{\Gamma _1}(X,Y)-1\Bigr ) Y^{n_2} + Y^{n_1} \Bigl (\mathcal {C}_{\Gamma _2}(X,Y)-1\Bigr ) \\ &\quad + {{\left(\sum _{\substack{\emptyset \not= U_1\subset V_1\\ \emptyset \not= U_2\subset V_2}} (X-1)^{\vert U_1\vert } (X-1)^{\vert U_2\vert }\right)}} Y^{n_1 + n_2-1}.\\[-42pt] \end{align*}□$\Box$As we will explain in Section 8.2, Proposition 5.2 is closely related to [29, Prop. 8.4].5.3ExampleComplete bipartite graphsLet Ka,b=Δa∨Δb$\operatorname{K}_{a,b} = \Delta _a \vee \Delta _b$ be a complete bipartite graph. Recall from Example 1.7 that CΔn(X,Y)=Xn$\mathcal {C}_{\Delta _n}(X,Y) = X^n$. Therefore, by Proposition 5.2, CKa,b(X,Y)=1+(Xa−1)Yb+Ya(Xb−1)+(Xa−1)(Xb−1)Ya+b−1$\mathcal {C}_{\operatorname{K}_{a,b}}(X,Y) = 1 + (X^a-1)Y^b + Y^a(X^b-1) + (X^a-1)(X^b-1)Y^{a+b-1}$. Hence,FKa,b(X,Y)=X(a−1)(b−1)(Xa−1)(Xb−1)Ya+b−1=+X(a−1)b(Xa−1)Yb+Xa(b−1)(Xb−1)Ya+Xab\begin{align*} \mathsf {F}_{\operatorname{K}_{a,b}}(X,Y) & = X^{(a-1)(b-1)}(X^a-1)(X^b-1)Y^{a+b-1} \\ & \phantom{=} +X^{(a-1)b}(X^a-1)Y^b + X^{a(b-1)}(X^b-1)Y^a +X^{ab} \end{align*}and fKa,b(X)=X(a−1)(b−1)((Xa−1)(Xb−1)+Xa−1(Xa−1)+Xb−1(Xb−1)+Xa+b−1)$f_{\operatorname{K}_{a,b}}(X) = X^{(a-1)(b-1)}((X^a-1)(X^b-1) + X^{a-1}(X^a-1) + X^{b-1}(X^b-1)+X^{a+b-1})$. We note that the graphical group GKa,b(Z/NZ)$\mathbf {G}_{\operatorname{K}_{a,b}}(\mathbf {Z}/N\mathbf {Z})$ is the maximal quotient of class at most 2 of the free product (Z/NZ)a∗(Z/NZ)b$(\mathbf {Z}/N\mathbf {Z})^a * (\mathbf {Z}/N\mathbf {Z})^b$; see [29, Section 3.4].5.4ExampleStarsAs a special case of Example 5.3, let Starn=Δn∨•=Kn,1$\operatorname{Star}_n = \Delta _{n} \vee \, \bullet = \operatorname{K}_{n,1}$ be a star graph on n+1$n+1$ vertices. Then CStarn(X,Y)=(Xn+1−Xn)Yn+(Xn−1)Y+1$\mathcal {C}_{\operatorname{Star}_n}(X,Y) = (X^{n+1}-X^n)Y^n + (X^n-1)Y + 1$. Hence, FStarn(X,Y)=Xn−1·(X2−X)Yn+(Xn−1)Y+X$\mathsf {F}_{\operatorname{Star}_n}(X,Y) = X^{n-1} \cdot \bigl ((X^2-X) Y^n + (X^n-1)Y + X\bigr )$ and fStarn(X)=Xn−1(Xn+X2−1)$f_{\operatorname{Star}_n}(X) =X^{n-1}(X^n+X^2-1)$.We record the following consequence of Proposition 5.2 for later use.5.5CorollaryLet Γ1$\Gamma _1$ and Γ2$\Gamma _2$ be graphs. Let Γi$\Gamma _i$ have mi$m_i$ edges and ni$n_i$ vertices. Then5.1fΓ1∨Γ2(X)=Xm1+m2+n1n2+Xm2+(n1−1)n2(fΓ1(X)−Xm1)+Xm1+n1(n2−1)(fΓ2(X)−Xm2)+Xm1+m2+(n1−1)(n2−1)(Xn1−1)(Xn2−1).\begin{align} f_{\Gamma _1\vee \Gamma _2}(X) &= X^{m_1 + m_2 + n_1 n_2} + X^{m_2 + (n_1-1)n_2} (f_{\Gamma _1}(X)-X^{m_1}) \nonumber \\ &\quad + X^{m_1 + n_1(n_2-1)} (f_{\Gamma _2}(X)-X^{m_2})\nonumber \\ &\quad + X^{m_1 + m_2 + (n_1-1)(n_2-1)}(X^{n_1}-1)(X^{n_2}-1). \end{align}Beyond disjoint unions and joins, it would be natural to study the effects of other graph operations on the polynomials CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$.THE CONSTANT AND LEADING TERM OF CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$Let Γ=(V,E)$\Gamma = (V,E)$ be a graph with n$n$ vertices and m$m$ edges. In this section, we primarily view CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ as a polynomial in Y$Y$ over Z[X]$\mathbf {Z}[X]$. Its constant term is easily determined.6.1PropositionCΓ(X,0)=Xi$\mathcal {C}_{\Gamma} (X,0) = X^i$, where i$i$ is the number of isolated vertices of Γ$\Gamma$.ProofAs C•(X,Y)=X$\mathcal {C}_\bullet (X,Y) = X$, by Proposition 5.1, CΓ⊕•(X,Y)=X·CΓ(X,Y)$\mathcal {C}_{\Gamma \oplus \bullet }(X,Y) = X\cdot \mathcal {C}_{\Gamma} (X,Y)$. We may thus assume that i=0$i = 0$. Let U⊂V$U\subset V$. As cΓ(U)⩽|U|⩽|NΓ[U]|$\operatorname{c}_{\Gamma} (U) \leqslant \vert U\vert \leqslant \vert \operatorname{N}_{\Gamma} [U]\vert$, we see that |NΓ[U]|=cΓ(U)$\vert \operatorname{N}_{\Gamma} [U]\vert = \operatorname{c}_{\Gamma} (U)$ if and only if U$U$ consists of isolated vertices. This only happens for U=∅$U = \emptyset$ whence CΓ(X,0)=1$\mathcal {C}_{\Gamma} (X,0) = 1$.□$\Box$For a group‐theoretic interpretation of Proposition 6.1, note that qi$q^i$ is the order of the quotient Z(GΓ(Fq))/[GΓ(Fq),GΓ(Fq)]$\operatorname{Z}(\mathbf {G}_{\Gamma} (\mathbf {F}_q))/[\mathbf {G}_{\Gamma} (\mathbf {F}_q),\mathbf {G}_{\Gamma} (\mathbf {F}_q)]$.Recall that rk(Γ)=n−c$\operatorname{rk}(\Gamma ) = n -c$, where c$c$ is the number of connected components of Γ$\Gamma$.6.2PropositiondegY(CΓ(X,Y))=rk(Γ)$\deg _Y(\mathcal {C}_{\Gamma} (X,Y)) = \operatorname{rk}(\Gamma )$.ProofIf Γ′$\Gamma ^{\prime }$ is any subgraph of Γ$\Gamma$, then rk(Γ′)⩽rk(Γ)$\operatorname{rk}(\Gamma ^{\prime }) \leqslant \operatorname{rk}(\Gamma )$. Let U⊂V$U\subset V$ and write U¯=NΓ[U]$\bar{U} = \operatorname{N}_{\Gamma} [U]$. Since every vertex in U¯∖U$\bar{U} \setminus U$ is adjacent to some vertex in U$U$, we have cΓ(U¯)⩽cΓ(U)$\operatorname{c}_{\Gamma} (\bar{U}) \leqslant \operatorname{c}_{\Gamma} (U)$. Hence, |U¯|−cΓ(U)⩽|U¯|−cΓ(U¯)=rk(Γ[U¯])⩽rk(Γ)$\vert \bar{U}\vert - \operatorname{c}_{\Gamma} (U) \leqslant \vert \bar{U}\vert - \operatorname{c}_{\Gamma} (\bar{U}) = \operatorname{rk}(\Gamma [\bar{U}]) \leqslant \operatorname{rk}(\Gamma )$. Thus, degY(CΓ(X,Y))⩽rk(Γ)$\deg _Y(\mathcal {C}_{\Gamma} (X,Y)) \leqslant \operatorname{rk}(\Gamma )$. The summand corresponding to U=V$U = V$ in (1.2) contributes a term XnYrk(Γ)$X^n Y^{\operatorname{rk}(\Gamma )}$ to CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$, and this term cannot be cancelled by a summand arising from any proper subset.□$\Box$For h(X,Y)=∑ijaijXiYj∈Z[X,Y]$h(X,Y) = \sum _{ij} a_{ij} X^iY^j\in \mathbf {Z}[X,Y]$ with aij∈Z$a_{ij}\in \mathbf {Z}$, write h(X,Y)Yj=∑iaijXi$h(X,Y)\Bigl [Y^j\Bigr ] = \sum _{i} a_{ij}X^i$ for the coefficient of Yj$Y^j$ in h(X,Y)$h(X,Y)$, regarded as a polynomial in Y$Y$. We now consider the leading coefficient CΓ(X,Y)Yrk(Γ)$\mathcal {C}_{\Gamma} (X,Y)\Bigl [Y^{\operatorname{rk}(\Gamma )}\Bigr ]$ of CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ as a polynomial in Y$Y$. Recall that a dominating set of Γ$\Gamma$ is a set D⊂V$D\subset V$ with NΓ[D]=V$\operatorname{N}_{\Gamma} [D] = V$. If, in addition, Γ[D]$\Gamma [D]$ is connected, then D$D$ is a connected dominating set. Let Dc(Γ)$\mathfrak {D}^{\mathrm{c}}(\Gamma )$ be the set of connected dominating sets of Γ$\Gamma$. Clearly, Dc(Γ)≠∅$\mathfrak {D}^{\mathrm{c}}(\Gamma ) \not= \emptyset$ if and only if Γ$\Gamma$ is connected.6.3PropositionSuppose that n⩾2$n\geqslant 2$. Then6.1CΓ(X+1,Y)Yn−1=∑D∈Dc(Γ)X|D|.\begin{equation} \mathcal {C}_{\Gamma} (X+1,Y)\Bigl [Y^{n-1} \Bigr ] = \sum \limits _{D \in \mathfrak {D}^{\mathrm{c}}(\Gamma )} X^{\vert D\vert }. \end{equation}ProofLet U⊂V$U\subset V$. As n⩾2$n \geqslant 2$, |NΓ[U]|−cΓ(U)=n−1$\vert \operatorname{N}_{\Gamma} [U]\vert - \operatorname{c}_{\Gamma} (U) = n-1$ if and only if NΓ[U]=V$\operatorname{N}_{\Gamma} [U] = V$ and cΓ(U)=1$\operatorname{c}_{\Gamma} (U) = 1$. The latter two conditions are satisfied if and only if U∈Dc(Γ)$U\in \mathfrak {D}^{\mathrm{c}}(\Gamma )$.□$\Box$6.4RemarkIn [23], the right‐hand side of (6.1) is referred to as the connected domination polynomial of Γ$\Gamma$. These polynomials are relatives of the widely studied domination polynomials of graphs introduced in [1] (where they were called dominating polynomials).6.5CorollarySuppose that Γ$\Gamma$ does not contain isolated vertices. Let V1,…,Vc⊂V$V_1,\ldots\,,V_c\subset V$ be the distinct connected components of Γ$\Gamma$. ThenCΓ(X+1,Y)Yrk(Γ)=∏i=1c∑Di∈Dc(Γ[Vi])X|Di|.\begin{equation*} \mathcal {C}_{\Gamma} (X+1,Y)\Bigl [Y^{\operatorname{rk}(\Gamma )} \Bigr ] = \prod _{i=1}^c \sum \limits _{D_i \in \mathfrak {D}^{\mathrm{c}}(\Gamma [V_i])} X^{\vert D_i\vert }. \end{equation*}The following is well known.6.6Theorem[9, Section A1.1, [GT2]]The problem of deciding, for a given graph Γ$\Gamma$ and k⩾1$k\geqslant 1$, whether Γ$\Gamma$ admits a connected dominating set of cardinality at most k$k$ is NP‐complete.DProof of PropositionCombine Theorem 6.6 and Proposition 6.3.□$\Box$We finish this section by showing that typically FΓ(0,Y)=0$\mathsf {F}_{\Gamma} (0,Y) = 0$. We first record the following consequence of Proposition 6.3:6.7CorollaryLet Γ$\Gamma$ be a tree with n⩾3$n\geqslant 3$ vertices and ℓ$\ell$ leaves. Then CΓ(X,Y)Yn−1=(X−1)n−ℓXℓ$\mathcal {C}_{\Gamma} (X,Y)\Bigl [Y^{n-1}\Bigr ] = (X-1)^{n-\ell } X^\ell$.ProofLet be the set of leaves of Γ$\Gamma$. Using n⩾3$n\geqslant 3$, it is easy to see that . Hence, by Proposition 6.3, .□$\Box$6.8CorollaryLet Γ$\Gamma$ be an arbitrary graph. Then FΓ(0,Y)=0$\mathsf {F}_{\Gamma} (0,Y) = 0$ unless Γ≈K2⊕r$\Gamma \approx \operatorname{K}_2^{\oplus r}$, in which case FΓ(0,Y)=(−1)rYr$\mathsf {F}_{\Gamma} (0,Y) = (-1)^rY^r$.ProofUsing Proposition 5.1 (and its evident analogue for FΓ(X,Y)$\mathsf {F}_{\Gamma} (X,Y)$), we may assume that Γ$\Gamma$ is connected. If m>rk(Γ)$m &gt; \operatorname{rk}(\Gamma )$, then X$X$ divides FΓ(X,Y)$\mathsf {F}_{\Gamma} (X,Y)$ by Proposition 6.2. Thus, suppose that m=rk(Γ)=n−1$m = \operatorname{rk}(\Gamma ) = n-1$, that is, Γ$\Gamma$ is a tree. Since FK1(X,Y)=X$\mathsf {F}_{\operatorname{K}_1}(X,Y) = X$ and FK2(X,Y)=(X2−1)Y+X$\mathsf {F}_{\operatorname{K}_2}(X,Y) = (X^2-1)Y + X$, we may assume that n⩾3$n \geqslant 3$. Corollary 6.7 then implies that FΓ(0,Y)=0$\mathsf {F}_{\Gamma} (0,Y) = 0$.□$\Box$6.9RemarkThe constant term and leading coefficient of CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ as a polynomial in X−1$X-1$ are easily determined: CΓ(1,Y)=1$\mathcal {C}_{\Gamma} (1,Y) = 1$ and CΓ(X+1,Y)=XnYrk(Γ)+O(Xn−1)$\mathcal {C}_{\Gamma} (X+1,Y) = X^n Y^{\operatorname{rk}(\Gamma )} + \mathcal {O}(X^{n-1})$. The constant term of CΓ(X,Y)$\mathcal {C}_{\Gamma} (X,Y)$ in X$X$, that is, the polynomial CΓ(0,Y)=∑U⊂V(−1)|U|Y|NΓ[U]|−cΓ(U)$\mathcal {C}_{\Gamma} (0,Y) = \sum _{U\subset V}(-1)^{\vert U\vert } Y^{\vert \operatorname{N}_{\Gamma} [U]\vert -\operatorname{c}_{\Gamma} (U)}$, seems to be more mysterious.THE DEGREES OF CLASS‐COUNTING POLYNOMIALSIn this section, we consider the degrees of class‐counting polynomials fΓ(X)=FΓ(X,1)$f_{\Gamma} (X) = \mathsf {F}_{\Gamma} (X,1)$ (see Theorem 1.2 and Corollary B). As before, let Γ=(V,E)$\Gamma = (V,E)$ be a graph with m$m$ edges and n$n$ vertices.Interpreting deg(fΓ(X))$\deg (f_{\Gamma} (X))$: The invariant η(Γ)$\eta (\Gamma )$For U⊂V$U\subset V$, let dΓ(U)=|NΓ[U]∖U|$\operatorname{d}_{\Gamma} (U) = \vert \operatorname{N}_{\Gamma} [U]\setminus U\vert$, the number of vertices in V∖U$V\setminus U$ with a neighbour in U$U$. Recall that cΓ(U)$\operatorname{c}_{\Gamma} (U)$ denotes the number of connected components of Γ[U]$\Gamma [U]$. Define7.1η(Γ)=maxU⊂VcΓ(U)−dΓ(U)⩾0.\begin{equation} \eta (\Gamma ) = \max _{U\subset V}\Bigl (\operatorname{c}_{\Gamma} (U) - \operatorname{d}_{\Gamma} (U)\Bigr ) \geqslant 0. \end{equation}Corollary B implies7.2deg(fΓ(X))=m+η(Γ).\begin{equation} \deg (f_{\Gamma} (X)) = m + \eta (\Gamma ). \end{equation}Our proof of Proposition D does not imply that computing fΓ(X)=XmCΓ(X,X−1)$f_{\Gamma} (X) = X^m \mathcal {C}_{\Gamma} (X,X^{-1})$ is NP‐hard, motivating Question 1.9.7.1QuestionIs there a polynomial‐time algorithm for computing η(Γ)$\eta (\Gamma )$?7.2RemarkThe author is unaware of previous investigations of the numbers η(Γ)$\eta (\Gamma )$ in the literature. At a formal level, η(Γ)$\eta (\Gamma )$ is reminiscent of other graph‐theoretic invariants such as critical independence numbers [37] (which can be computed in polynomial time).In the following, we establish bounds for η(Γ)$\eta (\Gamma )$. Let α(Γ)$\alpha (\Gamma )$ denote the independence number of Γ$\Gamma$, that is, the maximal cardinality of an independent set of vertices. Clearly,7.3η(Γ)⩽maxU⊂VcΓ(U)=α(Γ).\begin{equation} \eta (\Gamma ) \leqslant \max _{U\subset V} \operatorname{c}_{\Gamma} (U) = \alpha (\Gamma ). \end{equation}While η(Γ)$\eta (\Gamma )$ can be much smaller than α(Γ)$\alpha (\Gamma )$ (cf. Proposition 7.5(ii)), the bound η(Γ)⩽α(Γ)$\eta (\Gamma ) \leqslant \alpha (\Gamma )$ will be useful in our proof of Proposition 7.8 below.Let c$c$ be the number of connected components of Γ$\Gamma$. The case U=V$U = V$ in (7.1) shows that η(Γ)⩾c$\eta (\Gamma ) \geqslant c$. Since η(Γ1⊕Γ2)=η(Γ1)+η(Γ2)$\eta (\Gamma _1\oplus \Gamma _2) = \eta (\Gamma _1) + \eta (\Gamma _2)$, we may assume that Γ$\Gamma$ is connected.7.3PropositionLet Γ$\Gamma$ be connected and n⩾4$n\geqslant 4$. Then η(Γ)⩽n−2$\eta (\Gamma ) \leqslant n-2$ with equality if and only if Γ≈Starn−1$\Gamma \approx \operatorname{Star}_{n-1}$.ProofFor U∈{∅,V}$U\in \lbrace \emptyset , V\rbrace$, we have cΓ(U)−dΓ(U)⩽1<n−2$\operatorname{c}_{\Gamma} (U) - \operatorname{d}_{\Gamma} (U) \leqslant 1 &lt; n-2$. Let U⊂V$U\subset V$ with ∅≠U≠V$\emptyset \not= U\not= V$. Then cΓ(U)⩽n−1$\operatorname{c}_{\Gamma} (U) \leqslant n - 1$ and dΓ(U)>0$\operatorname{d}_{\Gamma} (U) &gt; 0$ since Γ$\Gamma$ is connected. Hence, cΓ(U)−dΓ(U)⩽n−2$\operatorname{c}_{\Gamma} (U) - \operatorname{d}_{\Gamma} (U) \leqslant n-2$ and η(Γ)⩽n−2$\eta (\Gamma ) \leqslant n-2$. Moreover, if cΓ(U)−dΓ(U)=n−2$\operatorname{c}_{\Gamma} (U) - \operatorname{d}_{\Gamma} (U) = n-2$, then cΓ(U)=n−1$\operatorname{c}_{\Gamma} (U) = n-1$ and dΓ(U)=1$\operatorname{d}_{\Gamma} (U) = 1$. This is equivalent to Γ$\Gamma$ being a star graph whose centre is the unique vertex in V∖U$V\setminus U$.□$\Box$By Proposition 7.3, η(Γ)$\eta (\Gamma )$ rarely attains its maximal value among graphs with n$n$ vertices. In contrast, η(Γ)=1$\eta (\Gamma ) = 1$ occurs frequently. Note that η(Kn)=1$\eta (\operatorname{K}_n) = 1$ by Example 1.3. For complete bipartite graphs, we obtain the following:7.4Propositionη(Ka,b)=max(1,|a−b|)$\eta (\operatorname{K}_{a,b}) = \max (1,\vert {a-b}\vert )$.ProofThis follows by inspection from the formula for fKa,b(X)$f_{\operatorname{K}_{a,b}}(X)$ in Example 5.3.□$\Box$Hence, η(Ka,b)=1$\eta (\operatorname{K}_{a,b}) = 1$ if and only if |a−b|⩽1$\vert {a-b}\vert \leqslant 1$. To obtain further examples of graphs Γ$\Gamma$ with η(Γ)=1$\eta (\Gamma ) = 1$, recall that a graph is claw‐free if it does not contain K1,3≈Star3$\operatorname{K}_{1,3} \approx \operatorname{Star}_3$ as an induced subgraph. The following proposition and its proof are due to Matteo Cavaleri. The author thanks him for kindly permitting this material to be included here.7.5Proposition   (i)η(Γ)=maxcΓ(U)+|U|−|V|:U⊂VisadominatingsetofΓ$\eta (\Gamma ) = \max \Bigl (\operatorname{c}_{\Gamma} (U) + \vert U\vert - \vert V\vert : U\subset V\text{ is a dominating set of } \Gamma \Bigr )$.(ii)If Γ$\Gamma$ is claw‐free and connected, then η(Γ)=1$\eta (\Gamma ) = 1$ and thus deg(fΓ(X))=m+1$\deg (f_{\Gamma} (X)) = m + 1$.Proof   (i)Let U⊂V$U\subset V$ with NΓ[U]≠V$\operatorname{N}_{\Gamma} [U]\not= V$. Let C⊂V$C\subset V$ be a connected component of Γ[V∖NΓ[U]]$\Gamma [V\setminus \operatorname{N}_{\Gamma} [U]]$. Clearly, cΓ(U∪C)=cΓ(U)+1$\operatorname{c}_{\Gamma} (U\cup C) = \operatorname{c}_{\Gamma} (U) + 1$. Let x∈NΓ[U∪C]∖(U∪C)$x\in \operatorname{N}_{\Gamma} [U\cup C]\setminus (U\cup C)$. Then x∼y$x\sim y$ for some y∈U∪C$y\in U \cup C$. Suppose that x∉NΓ[U]$x\not\in \operatorname{N}_{\Gamma} [U]$ so that y∈C$y\in C$. Then x∈C$x\in C$ by the definition of C$C$. This contradiction shows that NΓ[U∪C]∖(U∪C)⊂NΓ[U]∖U$\operatorname{N}_{\Gamma} [U\cup C]\setminus (U\cup C) \subset \operatorname{N}_{\Gamma} [U]\setminus U$. Hence, cΓ(U∪C)−dΓ(U∪C)>cΓ(U)−dΓ(U)$\operatorname{c}_{\Gamma} (U\cup C) - \operatorname{d}_{\Gamma} (U\cup C) &gt; \operatorname{c}_{\Gamma} (U) - \operatorname{d}_{\Gamma} (U)$. It follows that the maximal value of cΓ(U)−dΓ(U)$\operatorname{c}_{\Gamma} (U)-\operatorname{d}_{\Gamma} (U)$ is attained for a dominating set U$U$; in that case, dΓ(U)=|V|−|U|$\operatorname{d}_{\Gamma} (U) = \vert V\vert - \vert U\vert$.(ii)Let U⊂V$U\subset V$ be a ⊂$\subset$‐maximal dominating set with cΓ(U)+|U|−|V|=η(Γ)$\operatorname{c}_{\Gamma} (U) + \vert U\vert - \vert V\vert = \eta (\Gamma )$. Suppose that U≠V$U\not= V$. Choose x∈V∖U$x\in V\setminus U$. By maximality of U$U$, cΓ(U)+|U|>cΓ(U∪{x})+|U|+1$\operatorname{c}_{\Gamma} (U) + \vert U\vert &gt; \operatorname{c}_{\Gamma} (U\cup \lbrace x\rbrace ) + \vert U\vert + 1$. Hence, there are distinct connected components C1,C2,C3$C_1,C_2,C_3$ of Γ[U]$\Gamma [U]$ such that Γ[C1∪C2∪C3∪{x}]$\Gamma [C_1\cup C_2\cup C_3\cup \lbrace x\rbrace ]$ is connected. Choose ci∈Ci$c_i\in C_i$ with x∼ci$x\sim c_i$. Then Γ[{c1,c2,c3,x}]≈Star3$\Gamma [\lbrace c_1,c_2,c_3,x\rbrace ] \approx \operatorname{Star}_3$. We conclude that if Γ$\Gamma$ is claw‐free and connected, then U=V$U=V$ and thus η(Γ)=1$\eta (\Gamma ) = 1$.□$\Box$Let Δ(Γ)$\Delta (\Gamma )$ denote the maximum vertex degree of Γ$\Gamma$.7.6LemmaLet T$\mathsf {T}$ be a tree. Then η(T)⩾Δ(T)−1$\eta (\mathsf {T}) \geqslant \Delta (\mathsf {T})-1$.ProofLet w1,…,wd$w_1,\ldots\,,w_d$ be the distinct vertices adjacent to a vertex u$u$ of T$\mathsf {T}$. Let Wi$W_i$ consist of wi$w_i$ and all its descendants in the rooted tree (T,u)$(\mathsf {T},u)$. Define W:=W1∪⋯∪Wd$W := W_1\cup \cdots \cup W_d$. By construction, cT(W)=d$\operatorname{c}_\mathsf {T}(W) = d$ and dT(W)=1$\operatorname{d}_\mathsf {T}(W) = 1$ whence η(T)⩾d−1$\eta (\mathsf {T}) \geqslant d - 1$.□$\Box$7.7CorollaryLet T$\mathsf {T}$ be a tree. Then η(T)=1$\eta (\mathsf {T}) = 1$ if and only if T$\mathsf {T}$ is a path.ProofBy Lemma 7.6, η(T)>1$\eta (\mathsf {T}) &gt; 1$ unless T$\mathsf {T}$ is a path. By Proposition 7.5(ii) or Equation (1.1), we have η(Pn)=1$\eta (\operatorname{P}_{{n}}) = 1$.□$\Box$Upper and lower bounds for deg(fΓ(X))$\deg (f_{\Gamma} (X))$We obtain sharp bounds for deg(fΓ(X))$\deg (f_{\Gamma} (X))$ as Γ$\Gamma$ ranges over all graphs with n$n$ vertices.7.8PropositionLet Γ$\Gamma$ be a graph with n⩾1$n \geqslant 1$ vertices. Then n⩽deg(fΓ(X))⩽n2+1$n \leqslant \deg (f_{\Gamma} (X)) \leqslant \binom{n}{2} + 1$. The lower bound is attained if and only if Γ$\Gamma$ is a disjoint union of paths. The upper bound is attained if and only if Γ$\Gamma$ is complete or n=2$n = 2$.Our proof of Proposition 7.8 will rely on an upper bound for independence numbers.7.9Lemma[12]Let Γ$\Gamma$ be a graph with m$m$ edges and n$n$ vertices. Thenα(Γ)⩽12+14+n2−n−2m.\begin{equation*} \alpha (\Gamma ) \leqslant \left\lfloor \frac{1}{2} + \sqrt {\frac{1}{4} + n^2 - n-2m} \right\rfloor . \end{equation*}7.8Proof of PropositionAs before, let m$m$ denote the number of edges of Γ=(V,E)$\Gamma = (V,E)$.(i)Lower bound.We may assume that Γ$\Gamma$ is connected so that m⩾n−1$m\geqslant n-1$. As η(Γ)⩾1$\eta (\Gamma ) \geqslant 1$, Equation (7.2) shows that deg(fΓ(X))⩾n$\deg (f_{\Gamma} (X)) \geqslant n$ with equality if and only if Γ$\Gamma$ is a tree and η(Γ)=1$\eta (\Gamma ) = 1$. By Corollary 7.7, the latter condition is equivalent to Γ≈Pn$\Gamma \approx \operatorname{P}_{{n}}$.(ii)Upper bound.Since deg(fKn(X))=n2+1$\deg (f_{\operatorname{K}_n}(X)) = \binom{n}{2} + 1$ by Example 1.3, it suffices to show that deg(fΓ(X))⩽n2$\deg (f_{\Gamma} (X)) \leqslant \binom{n}{2}$ whenever m<n2$m &lt; \binom{n}{2}$. We may assume that n⩾3$n\geqslant 3$. Writing m=n2−k$m = \binom{n}{2} - k$, Lemma 7.9 shows that α(Γ)⩽⌊12+2k+14⌋$\alpha (\Gamma ) \leqslant \lfloor \frac{1}{2} + \sqrt {2k+\frac{1}{4}} \rfloor$. Hence, if k⩾2$k\geqslant 2$, then α(Γ)⩽k$\alpha (\Gamma ) \leqslant k$. By Equation (7.3), deg(fΓ(X))=m+η(Γ)⩽m+α(Γ)⩽m+k=n2$\deg (f_{\Gamma} (X)) = m + \eta (\Gamma ) \leqslant m + \alpha (\Gamma ) \leqslant m + k = \binom{n}{2}$. For k=1$k = 1$, we have Γ≈Δ2∨Kn−2$\Gamma \approx \Delta _2 \vee \operatorname{K}_{n-2}$ and deg(fΓ(X))=n2$\deg (f_{\Gamma }(X)) = \binom{n}{2}$ by Proposition 7.5(ii). (Alternatively, we may combine Corollary 5.5 and Example 1.3.)□$\Box$APPLICATIONS TO ZETA FUNCTIONS OF GRAPHICAL GROUP SCHEMESWe briefly relate some of our findings to recent work on zeta functions of groups.Reminder: Class‐counting and conjugacy class zeta functionsThe study of zeta functions associated with groups and group‐theoretic counting problems goes back to the influential work of Grunewald et al. [11]. Let G$\mathbf {G}$ be a group scheme of finite type over a compact discrete valuation ring O$\mathfrak {O}$ with maximal ideal P$\mathfrak {P}$. The class‐counting zeta function of G$\mathbf {G}$ is the Dirichlet series ζGk(s)=∑i=0∞k(G(O/Pi))|O/Pi|−s$\zeta ^{\operatorname{k}}_{\mathbf {G}}(s) = \sum _{i=0}^\infty \operatorname{k}(\mathbf {G}(\mathfrak {O}/\mathfrak {P}^i)) {\vert \mathfrak {O}/\mathfrak {P}^i\vert} ^{-s}$. Beginning with work of du Sautoy [7], these and closely related series enumerating conjugacy classes have recently been studied, see [3, 19, 20, 27–29]. Recall the definition of the conjugacy class zeta function ζGcc(s)$\zeta ^{\rm cc}_G(s)$ associated with a finite group G$G$ from Section 4. Lins [19, Def. 1.2] introduced a refinement of ζGk(s)$\zeta ^{\operatorname{k}}_{\mathbf {G}}(s)$, the bivariate conjugacy class zeta function ζGcc(s1,s2)=∑i=0∞ζG(O/Pi)cc(s1)|O/Pi|−s2$\zeta ^{\rm cc}_{\mathbf {G}}(s_1,s_2) =\sum _{i=0}^\infty \zeta _{\mathbf {G}(\mathfrak {O}/\mathfrak {P}^i)}^{\rm cc}(s_1) \vert \mathfrak {O}/\mathfrak {P}^i\vert ^{-s_2}$ of G$\mathbf {G}$ and studied these functions for certain classes of unipotent group schemes; note that ζGk(s)=ζGcc(0,s)$\zeta ^{\operatorname{k}}_{\mathbf {G}}(s) = \zeta ^{\rm cc}_{\mathbf {G}}(0,s)$.Theorem 1.2 is in fact a special case of a far more general result pertaining to class‐counting zeta functions associated with graphical group schemes.8.1TheoremCf. [29, Cor. B]For each graph Γ$\Gamma$, there exists a rational function W∼Γ(X,Y)∈Q(X,Y)${\tilde{W}}_{\Gamma} (X,Y) \in \mathbf {Q}(X,Y)$ with the following property: For each compact discrete valuation ring O$\mathfrak {O}$ with residue field size q$q$, we have ζGΓ⊗Ok(s)=W∼Γ(q,q−s)$\zeta ^{\operatorname{k}}_{\mathbf {G}_{\Gamma} \otimes \mathfrak {O}}(s)= {\tilde{W}}_{\Gamma} (q,q^{-s})$.Theorem 8.1 contains Theorem 1.2 as a special case via W∼Γ(X,Y)=1+fΓ(X)Y+O(Y2)$\tilde{W}_{\Gamma} (X,Y) = 1 + f_{\Gamma} (X) Y + \mathcal {O}(Y^2)$.8.2RemarkIn the present article, we chose to normalise our polynomials and rational functions slightly differently compared to [29]. Namely, what we call W∼Γ(X,Y)$\tilde{W}_{\Gamma} (X,Y)$ here coincides with WΓ−(X,XmY)$W_{\Gamma} ^-(X,X^mY)$ in [29], where m$m$ is the number of edges of Γ$\Gamma$.Class‐counting zeta functions of graphical group schemes and joinsLet Γ1$\Gamma _1$ and Γ2$\Gamma _2$ be graphs with n1$n_1$ and n2$n_2$ vertices and m1$m_1$ and m2$m_2$ edges, respectively. Define a rational function QΓ1,Γ2(X,Y)∈Q(X,Y)$Q_{\Gamma _1,\Gamma _2}(X,Y) \in \mathbf {Q}(X,Y)$ viaQΓ1,Γ2(X,Y)=Xm1+m2+(n1−1)(n2−1)Y−1=+W∼Γ1(X,Xm2+(n1−1)n2Y)·(1−Xm1+m2+(n1−1)n2Y)(1−Xm1+m2+(n1−1)n2+1Y)=+W∼Γ2(X,Xm1+n1(n2−1)Y)·(1−Xm1+m2+n1(n2−1)Y)(1−Xm1+m2+n1(n2−1)+1Y).\begin{align*} Q_{\Gamma _1,\Gamma _2}(X,Y) & = X^{m_1 + m_2 + (n_1-1)(n_2-1)}Y - 1 \\ & \phantom{=} \, + \tilde{W}_{\Gamma _1}(X,X^{m_2 + (n_1-1)n_2}Y)\cdot (1-X^{m_1 + m_2 + (n_1-1)n_2}Y) (1-X^{m_1 + m_2 + (n_1-1)n_2+1}Y)\\ & \phantom{=} \, + \tilde{W}_{\Gamma _2}(X,X^{m_1 + n_1(n_2-1)}Y)\cdot (1-X^{m_1 + m_2 + n_1(n_2-1)}Y) (1-X^{m_1 + m_2 + n_1(n_2-1)+1}Y). \end{align*}Our study of joins in Section 5 was motivated by the following:8.3Theorem[29, Prop. 8.4]Suppose that Γ1$\Gamma _1$ and Γ2$\Gamma _2$ are cographs. Then8.1W∼Γ1∨Γ2(X,Y)=QΓ1,Γ2(X,Y)(1−Xm1+m2+n1n2Y)(1−Xm1+m2+n1n2+1Y).\begin{equation} \tilde{W}_{\Gamma _1\vee \Gamma _2}(X,Y) = \frac{Q_{\Gamma _1,\Gamma _2}(X,Y)}{(1-X^{m_1+m_2+n_1n_2}Y)(1-X^{m_1+m_2+n_1n_2+1}Y)}. \end{equation}It remains unclear whether the assumption that Γ1$\Gamma _1$ and Γ2$\Gamma _2$ be cographs in Theorem 8.3 is truly needed or if it is merely an artefact of the proof given in [29].8.4Question[29, Question 10.1]Does (8.1) hold for arbitrary graphs Γ1$\Gamma _1$ and Γ2$\Gamma _2$?We obtain a positive answer to a (much weaker!) ‘approximate form’ of Question 8.4.8.5PropositionLet Γ1$\Gamma _1$ and Γ2$\Gamma _2$ be arbitrary graphs with n1$n_1$ and n2$n_2$ vertices and m1$m_1$ and m2$m_2$ edges, respectively. Then, regarded as formal power series in Y$Y$ over Q(X)$\mathbf {Q}(X)$, the rational function W∼Γ1∨Γ2(X,Y)$\tilde{W}_{\Gamma _1\vee \Gamma _2}(X,Y)$ and the right‐hand side of (8.1) agree modulo Y2$Y^2$.ProofThis follows from Corollary 5.5: by expanding the right‐hand side of (8.1) as a series in Y$Y$, we find that the coefficient of Y$Y$ is given by the right‐hand side of (5.1).□$\Box$Uniformity and the difficulty of computing zeta functions of groupsFor reasons that are not truly understood at present, many interesting examples of zeta functions associated with group‐theoretic counting problems are ‘(almost) uniform’. As we now recall, the task of symbolically computing such zeta functions is well defined.UniformityBeginning with a global object G$G$ and a type of counting problem, we often obtain (a) associated local objects Gp$G_p$ indexed by primes (or places) p$p$ and (b) associated local zeta functions ζGp(s)$\zeta _{G_p}(s)$. The family (ζGp(s))p$(\zeta _{G_p}(s))_p$ of zeta functions is (almost) uniform if there exists WG(X,Y)∈Q(X,Y)$W_G(X,Y)\in \mathbf {Q}(X,Y)$ such that ζGp(s)=WG(p,p−s)$\zeta _{G_p}(s) = W_G(p,p^{-s})$ for (almost) all p$p$. (Stronger forms of uniformity may also take into account local base extensions or changing the characteristic of compact discrete valuation rings under consideration. Variants apply to multivariate zeta functions such as ζGcc(s1,s2)$\zeta ^{\rm cc}_{\mathbf {G}}(s_1,s_2)$.) It is then natural to seek to devise algorithms for computing WG(X,Y)$W_G(X,Y)$ and to consider the complexity of such algorithms.Numerous computations of (almost) uniform zeta functions associated with groups and related algebraic structures have been recorded in the literature; see, for example, [8]. For a recent example, Carnevale et al. [5] (see also [6]) obtained strong uniformity results for ideal zeta functions of certain nilpotent Lie rings. Their explicit formulae for rational functions as sums over chain complexes involve sums of super‐exponentially many rational functions.The following example illustrates how class‐counting zeta functions associated with graphical group schemes fit the above template for uniformity of zeta functions.8.6ExampleLet G=GΓ$G = \mathbf {G}_{\Gamma}$ be a graphical group scheme. For a prime p$p$, let Zp$\mathbf {Z}_p$ denote the ring of p$p$‐adic integers and let Gp=G⊗Zp$G_p = G \otimes \mathbf {Z}_p$. Writing ζGp(s)=ζGpk(s)$\zeta _{G_p}(s) = \zeta ^{\operatorname{k}}_{G_p}(s)$, the family (ζGp(s))p$(\zeta _{G_p}(s))_p$ is uniform by Theorem 8.1 with WG(X,Y)=W∼Γ(X,Y)$W_G(X,Y) = \tilde{W}_{\Gamma }(X,Y)$. The constructive proof of Theorem 8.1 in [29] gives rise to an algorithm for computing W∼Γ(X,Y)$\tilde{W}_{\Gamma} (X,Y)$ (see [29, Section 9.1]). While no complexity analysis was carried out in [29], this algorithm appears likely to be substantially worse than polynomial‐time. For a cograph Γ$\Gamma$, [29, Thms C–D] combine to produce a formula for W∼Γ(X,Y)$\tilde{W}_{\Gamma} (X,Y)$ as a sum of explicit rational functions, the number of which grows super‐exponentially with the number of vertices of Γ$\Gamma$.Computing bivariate conjugacy class zeta functionsAs indicated (but not spelled out as such) in [29, Section 8.5], Theorem 8.1 admits the following generalisation: given a graph Γ$\Gamma$, there exists W∼Γ(X,Y,Z)∈Q(X,Y,Z)$\tilde{W}_{\Gamma} (X,Y,Z)\in \mathbf {Q}(X,Y,Z)$ such that for all compact discrete valuation rings with residue field size q$q$, ζGΓcc(s1,s2)=W∼Γ(q,q−s1,q−s2)$\zeta ^{\rm cc}_{\mathbf {G}_{\Gamma} }(s_1,s_2) = \tilde{W}_{\Gamma} (q,q^{-s_1},q^{-s_2})$. (Hence, W∼Γ(X,1,Z)=W∼Γ(X,Z)$\tilde{W}_{\Gamma} (X,1,Z) = \tilde{W}_{\Gamma} (X,Z)$.) Suppose that, given Γ$\Gamma$, an oracle provided us with W∼Γ(X,Y,Z)$\tilde{W}_{\Gamma} (X,Y,Z)$ as a reduced fraction of polynomials. Since W∼Γ(X,Y,Z)=1+FΓ(X,Y)Z+O(Z2)$\tilde{W}_{\Gamma} (X,Y,Z) = 1 + \mathsf {F}_{\Gamma} (X,Y) Z + \mathcal {O}(Z^2)$, we may then compute FΓ(X,Y)$\mathsf {F}_{\Gamma} (X,Y)$ by symbolic differentiation. In particular, Proposition D implies that computing W∼Γ(X,Y,Z)$\tilde{W}_{\Gamma} (X,Y,Z)$ is NP‐hard. To the author's knowledge, this is the first non‐trivial lower bound for the difficulty of computing uniform zeta functions associated with groups. We do not presently obtain a similar lower bound for the difficulty of computing W∼Γ(X,Y)$\tilde{W}_{\Gamma} (X,Y)$ since the difficulty of determining fΓ(X)$f_{\Gamma} (X)$ remained unresolved in Section 7.Open problem: Higher congruence levelsIt is an open problem to find a combinatorial formula for the rational functions W∼Γ(X,Y)$\tilde{W}_{\Gamma} (X,Y)$ (or their generalisations W∼Γ(X,Y,Z)$\tilde{W}_{\Gamma} (X,Y,Z)$ from Section 8.3) as Γ$\Gamma$ ranges over all graphs on a given vertex set; cf. [29, Question 1.8(iii)]. Corollary B provides such a formula for the first non‐trivial coefficient of W∼Γ(X,Y)=1+fΓ(X)Y+O(Y2)$\tilde{W}_{\Gamma} (X,Y) = 1 + f_{\Gamma} (X) Y + \mathcal {O}(Y^2)$, and Theorem A provides a formula for the first non‐trivial coefficient of W∼Γ(X,Y,Z)=1+FΓ(X,Y)Z+O(Z2)$\tilde{W}_{\Gamma} (X,Y,Z)= 1 + \mathsf {F}_{\Gamma} (X,Y) Z + \mathcal {O}(Z^2)$. As suggested by one of the anonymous referees, it is natural to ask whether a combinatorial formula of the type considered here can be obtained for the coefficient of Y2$Y^2$ in W∼Γ(X,Y)$\tilde{W}_{\Gamma} (X,Y)$ or of Z2$Z^2$ in W∼Γ(X,Y,Z)$\tilde{W}_{\Gamma} (X,Y,Z)$. These coefficients enumerate conjugacy classes of graphical groups GΓ(O/P2)$\mathbf {G}_{\Gamma} (\mathfrak {O}/\mathfrak {P}^2)$, where O$\mathfrak {O}$ is a compact discrete valuation ring with maximal ideal P$\mathfrak {P}$. The ‘dual’ problem of enumerating characters (see Section 1.6) is related to recent research developments. 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