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Optimizing Wiener and Randić Indices of Graphs

Optimizing Wiener and Randić Indices of Graphs Hindawi Advances in Operations Research Volume 2020, Article ID 3139867, 10 pages https://doi.org/10.1155/2020/3139867 Research Article Optimizing Wiener and Randic Indices of Graphs A. C. Mahasinghe , K. K. W. H. Erandi , and S. S. N. Perera Research & Development Center for Mathematical Modeling, Department of Mathematics, Faculty of Science, University of Colombo, Colombo 03, Sri Lanka Correspondence should be addressed to A. C. Mahasinghe; anuradhamahasinghe@gmail.com Received 9 June 2020; Accepted 4 August 2020; Published 26 September 2020 Academic Editor: Dylan F. Jones Copyright © 2020 A. C. Mahasinghe et al. %is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Wiener and Randi´ c indices have long been studied in chemical graph theory as connection strength measures of graphs. Later, these indices were used in different fields such as network analysis. We consider two optimization problems related to these indices, with potential applications to network theory, in particular to epidemiological networks. Given a connected graph and a fixed total edge weight, we investigate how individual weights must be assigned to edges, minimizing the connection strength of the graph. In order to measure the connection strength, we use the weighted Wiener index and a modified version of the ordinary ´ ´ Randic index. Wiener index optimization is linear, while Randic index optimization turns out to be both nonlinear and nonconvex. Hence, we adopt the technique of separable programming to generate solutions. We present our experimental results by applying relevant algorithms to several graphs. Mohar in 1996, in which it was proven that a variant of the 1. Introduction Wiener index coincides with the Kirchhoff index of electrical networks [5]. In 2002, Otte and Rousseau further extended Topological indices of graphs have served as numerical invariants of chemical structures, characterizing the to- the scope of topological indices by using them to analyze pology of the chemical structure graph theoretically. In social networks [6]. In a recent work, Imran et al. analyzed most cases, these indices were used to measure the interconnection networks using topological indices [7]. All connection strength of chemical compounds. %e first- these works used topological indices to characterize existing ever such topological index found in the literature was the networks with fixed vertex and edge weights in order to Wiener index, of which the intention was exploring derive information about the network. In contrast to this, thermodynamic and physiochemical properties of alkanes Ghosh et al. investigated how the edge weights can be in terms of molecular shapes [1].Consequently, variants of assigned subject to a fixed total weight, in order to optimize a the Wiener index and different other indices appeared for topological index, together with an application into electrical similar purposes, introducing a new field, chemical graph circuits [8]. %is application was interpreted as assigning resistors to the edges of an electric network, subject to a total theory, into theoretical chemistry [2–4]. %ough different indices intended for different characterizations of fixed sum of resistance, aimed at minimizing the total ef- chemical compounds, they shared in common the notion fective resistance or, analogously, maximizing the connec- of connection strength or compactness of the relevant tion strength. %e topological measure for the total effective graph structure. resistance has been taken as the algebraic connectivity %ough these indices were originally confined to the (smallest nontrivial Laplacian eigenvalue) of the relevant chemical graph theory, their scope has later been extended as graph, which is proven to be closely associated with the to include other subject areas as well. %e applicability of Wiener index [5]. Interestingly, the relevant optimization topological indices to networks beyond chemical structures problem turns out to be convex, guaranteeing efficient was initiated with the pioneering work by Gutman and solvability. 2 Advances in Operations Research the aim is the minimization of the compactness of the ep- Optimizing topological indices subject to different constraints in different contexts has been the subject of idemiological network, it is equivalent to optimizing an appropriate topological index subject to a fixed total edge interest in several previous works. Most of these optimi- zation problems were related to the chemical graph theory, weight. %is is the main problem of interest in this work. in particular to the design of chemical compounds. In [9], We first consider the Wiener index, which is the simplest Raman and Maranas developed an integer programming topological index for characterizing the compactness of the model to optimize a combination of topological indices network. However, in the context of epidemiological net- including Wiener and Randic ´ indices and Kier’s shape index works, a distance-based measure as Wiener index is less [10]. Optimizing the molecular interconnectivity index for significant than a degree-based measure as a region with many interconnections is likely to contribute significantly to the design of polymers had been discussed in [11], with a solution scheme for the resulting nonconvex mixed-integer the spread of the disease. We find the degree-based topo- logical index introduced by Randic ´ in 1975 [21] ideal for our linear programming formulation. A computational scheme for designing new molecules in medicinal chemistry was purpose. %ough this was originally intended for measuring the extent of branching of the carbon atom in hydrocarbons, described in [12] by Siddhaye et al., where the first-order molecular connectivity index was optimized through an similar to the Wiener index, later developments of the integer programming reformulation and the branch-and- Randic ´ index have proven its applicability in different bound approach. An optimization problem of a different contexts [22–24]. flavour in the context of chemical graph theory was con- Accordingly, we consider both Wiener index and Randic ´ sidered in [13], where the simplex algorithm was used to index to measure the compactness of the network in our derive optimal versions of several topological indices. optimization problem. %e problem of optimizing the Wiener index turns out to be linear, thus trivially solvable. A few works on optimizing topological indices appear outside the realm of chemical graph theory as well. A On the contrary, optimizing the Randic ´ index is a chal- lenging task as it turns out to be both nonlinear and non- concept paper by Preuß et al. [14] had proposed the opti- mization of Wiener and Randic ´ indices to solve the maxi- convex. In order to overcome the computational hardness, we adopt the technique of separable programming [25, 26] mum terrain coverage problem. A recent work [15] explored the possibility of optimizing the Wiener index for solving the and replace respective nonlinear functions by their piecewise critical node detection problem, where Benders algorithm linear approximations, eventually ending up with an ap- [16] was adopted as the solution technique. An application of proximate solution to the problem. algebraic connectivity maximization to communication %e remainder of the paper is organized as follows. In networks was discussed in [17]. Section 2, we reformulate the problem of minimizing the %e specific optimization problem considered by Ghosh Wiener index of a graph subject to a fixed total edge weight as a linear program. In Section 3, we consider the same et al. [8] is of particular interest and can be contrasted with the other optimization models with topological indices as optimization problem by replacing the Wiener index with the Randic ´ index, which turns out to be nonconvex. Our the optimization is done subject to a constant edge weight sum. It is natural to seek what the nature of the problem reformulation using separable programming techniques can would be if algebraic connectivity is replaced by a different be found in the same section. Section 4 contains our topological index. Would that be an efficiently solvable computational results, from which the discussion in Section optimization problem? Also, what if the objective was 5 is motivated. changed to minimizing the connection strength contrary to the electric network context? %ese are not merely questions 2. Optimizing the Wiener Index of theoretical interest; there is a useful application to epi- demiological networks. Consider the transmission of a %e simplest and the pioneering topological index of graphs vector-borne disease throughout a geographical region. %is is the Wiener index. Consider a simple connected undi- region can be considered as a network, of which the vertices rected graph G(V, E) with n vertices. %en, the Wiener index represent cities or suburbs, while the edges represent their W(G) of a graph G is defined as interconnections such as roads and channels, along which n n the vector-borne disease transmits [18]. %e weight of an W(G) � 􏽘 􏽘 dist(i, j), (1) edge in this network could be regarded as a measure of i�1 j�1 favorable conditions for breeding sites of vectors. It has been claimed that the rapid transmission of such a disease is where dist(i, j) is the distance between ith and jth vertices of largely influenced by the compactness of the network G. One may find several papers on finding Wiener indices of [19, 20]. %us, the health planners might be interested in different graphs [27–29]. %ough it is the unweighted ver- minimizing the compactness of this network by eliminating sion of the Wiener index which is used often, the vertex- the favorable conditions for vectors along the roads or water weighted graph version was introduced later [30], which has channels. However, this procedure is not without budgetary achieved progress in the past few years [31, 32]. Further- constraints. %e total amount of budget available in the more, an edge-weighted version (known as the Gutman control process for eliminating vector breeding sites must be index) was introduced as a natural extension of the Wiener optimally utilized along the roads and channels. %us, the index [33]. We follow this as the edge-weighted Wiener total edge weight must be bounded by a constant. Whenever index in our optimization problem. Accordingly, for a graph Advances in Operations Research 3 G together with functions d and w, where d is the degree of Aimed at minimizing the edge-weighted Randic index, the vertex i and w is the weight of the edge (i, j), the now we express the objective function to be minimized as ij weighted Wiener index is expressed as follows: Z � R (G), subject to the same budgetary and non- 2 w negativity constraints given by (3) and (4), respectively. %us, the problem turns out to be nonlinear. A closer W (G) � 􏽘 􏽘 w d d . (2) w ij i j i�1 look might reveal its nonconvexity. Recalling the problem of (i,j)∈E optimizing the topological index taken as algebraic con- Our objective is minimizing the function Z � W (G) in 1 w nectivity turned out to be convex, a closed-form solution was equation (2), subject to relevant constraints. In terms of the obtained [8]. In contrast to this, optimizing the Randic ´ index epidemiological network context, the sole major constraint is is a nonconvex optimization problem; thus, it is challenging the availability of budget for the control process. %is constraint to find the global optimum. Hence, instead of looking for enforces that the allocation of resources to roads and channels closed-form exact solutions, we seek an approximate solu- must be done subject to a fixed total budget. Let the total tion by using appropriate optimization and approximation (maximum possible) budget for resources be denoted by C. schemes. It is not difficult to see that our objective function %en, the relevant constraint can be expressed as in equation (6) can be easily convertible to a form expressible as sums of functions of individual decision variables. %is 􏽘 w ≤ C. motivates us to make use separable programming technique ij (3) (i,j)∈E for approximating the solution. Finally, the nonnegativity of w must be specified by ij 3.1. Separable Programming Formulation. %e method of ∀(i, j) ∈ E, w ≥ 0. (4) separable programming was first introduced for constrained ij optimization of nonlinear convex functions, whenever these It can be seen that the optimization problem given by functions are expressible as sums of functions of a single equations (2)–(4) is linear. %erefore, the solution is non- variable [25]. Functions with the latter mentioned property trivially obtained for Wiener index optimization. were called separable, and later works investigated the possibility of expanding the technique to nonconvex func- tions as well [42, 43]. Since its inception, separable pro- 3. Optimizing the Randic´ Index gramming has been a very useful optimization technique, with applications to several real problems including agri- %e topological index introduced for characterizing the con- cultural planning [44], linear complementarity problem nection strength of chemical compounds by Milan Randic ´ is [45], newsboy problem [46], and demand allocation [47]. widely known in chemical graph theory. %is has been a subject Notice that the objective function in equation (6) is of interest for graph theorists, and several works can be found nonlinear and nonconvex. Hence, we convert the objective on graph-theoretic aspects of the Randic ´ index [34–36]. Being function to a separable form. Let an elegant measure for the connection strength, this index has been applied in different contexts as well. Examples include measuring the robustness in cybernetics [22], reliability of a � 􏽘 w . (7) i ij communication networks [37], connectivity of mobile net- j�1 works [38], and information content of a graph [39]. %e ordinary Randic ´ index is expressed as follows: From equation (6), our objective function can be restated as 􏽰������� R(G) � 􏽘 . (5) √�� n 2 d(i)d(j) n n n (i,j)∈E 􏽐 a 􏼁 1 √��� � i�1 ⎝ ⎠ ⎛ ⎞ Z � � 􏽘 a + 􏽘 􏽘 a a . (8) 2 i i s C C i�1 i�1 s�1 Clearly, Randic ´ index is a degree-based topological in- dex, unlike the distance-based Wiener index. %us, unlike √�� √��� � Since a > 0 for all i ∈ {1, 2, . . . , n}, a a can be i i s Wiener index optimization which was more into theoretical replaced by y , where r ∈ {1, 2, . . . , n(n − 1)/2}. %en, it is interest, Randic ´ index optimization is directly relevant to our possible to transform the objective function to the separable epidemiological application mentioned in Section 1. form with the following constraint in the separable form: Moreover, a reduction of the Randic ´ index results in the √�� √�� reduction of the Wiener index, a fact which can be seen by log y � log a + log a . (9) 􏼁 􏼁 􏼁 r i s comparing equations (1) and (5) [40]. Now, the objective function can be restated as %e edge-weighted version of the Randic ´ index was introduced by Araujo and De la Peña as follows [41]: n ((n(n−1))/2) ⎛ ⎝ ⎞ ⎠ 􏽱�������� � Z � a + 2 􏽘 y . (10) 2 􏽘 2 i r 􏽐 􏽐 w 􏼒 􏼓 i�1 r�1 i�1 (i,j)∈E ij (6) R (G) � . 􏽐 􏽐 w %is is to be minimized subject to i�1 (i,j)∈E ij 4 Advances in Operations Research 􏽘 􏽘 w ≤ C, (11a) ij i�1 j∈V a � 􏽘 w , ∀i ∈ 1, 2, . . . , n , { } (11b) i ij j�1 1, 2, . . . , n(n − 1) √� � √�� log y 􏼁 � log a 􏼁 + log a 􏼁 , ∀r ∈ 􏼨 􏼩, (11c) r i s w ≥ 0, ∀i, j ∈ {1, 2, . . . , n}, (11d) ij a > 0, ∀i ∈ {1, 2, . . . , n}, (11e) 1, 2, . . . , n(n − 1) y > 0, ∀r ∈ 􏼨 􏼩. (11f ) 3.2. Linearly Approximated Program. Notice that the con- straint given by equation (11c) is nonlinear. In order to y � 􏽘 λ α , r rk rk k�1 approximate by piecewise linear functions, first, we restate (19) equation (11c) as 􏽘 λ � 1. 2 rk 1, 2, . . . , n(n − 1) k�1 g y 􏼁 + 􏽘 g 􏼐x 􏼑 � 0, ∀r ∈ 􏼨 􏼩, r1 r r(1+p) p p�1 Let the domain of g (x ) be the interval [C , C]. %en, rp p 1 (12) we divide the interval into h subdivisions of length q by defining β as follows: pl where g y 􏼁 � log y , (13a) C � β ≤ β ≤ · · · ≤ β � C, (20) r1 r r 0 p1 p2 ph √�� where g x 􏼁 � −log a , (13b) r2 1 i 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 √�� 􏼌 􏼌 β − β � q. (21) 􏼌 􏼌 pl p(l+1) g x 􏼁 � −log a . (13c) r3 2 s %en, the piecewise linear approximation to g (x ) can Let the domain of g (y ) be the interval [C , C]. %en, rp p r1 r 0 be expressed as we divide the interval into m subdivisions of length d by defining α as follows: rk (22) C � α ≤ α ≤ · · · ≤ α � C, (14) g 􏼐x 􏼑 � 􏽘 μ β , 0 r1 r2 rm rp p pl pl l�1 where where 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 α − α � d. (15) 􏼌 􏼌 rk r(k+1) x � 􏽘 μ β , p pl pl Now, a point y ∈ [C , C] can be uniquely expressed as r 0 l�1 (23) y � λ α + λ α � 1, (16) r rk rk r(k+1) r(k+1) h 􏽘 μ � 1. pl where l�1 λ + λ � 1. (17) rk r(k+1) Now, the nonlinear program is approximated by the following problem: %en, the piecewise linear approximation to g (y ) can r1 r ((n(n−1))/2) be expressed as m n h ⎝ ⎠ ⎛ ⎞ minimize Z � 􏽘 􏽘 λ α + 􏽘 􏽘 μ β , m 3 rk rk il il r�1 i�1 k�1 l�1 g y 􏼁 � 􏽘 λ g α 􏼁 , (18) r1 r rk r1 rk k�1 (24) where subject to Advances in Operations Research 5 􏽘 􏽘 w ≤ C, (25a) ij i�1 j∈V n h 􏽘 􏽘 μ β � 􏽘 w , (25b) il il ij i�1 l�1 (i,j)∈E m 2 h 1, 2, . . . , n(n − 1) 􏽘 λ g α 􏼁 + 􏽘 􏽘 μ g 􏼐β 􏼑 � 0, ∀r ∈ 􏼨 􏼩, (25c) rk r1 rk pl rp pl p�1 k�1 l�1 1, 2, . . . , n(n − 1) 􏽘 λ � 1, ∀r ∈ 􏼨 􏼩, (25d) rk k�1 1, 2, . . . , n(n − 1) λ , α ≥ 0, ∀r ∈ 􏼨 􏼩, ∀k ∈ {1, 2, . . . , m,} (25e) rk rk 􏽘 μ � 1, ∀i ∈ {1, 2, . . . , n}, (25f ) il l�1 μ , β ≥ 0, ∀i ∈ {1, 2, . . . , n}, ∀l ∈ {1, 2, . . . , h}, (25g) il il and at most two adjacent λ ’s are positive. equal to one, the optimal solution assigned a total weight of 1 rk It can be seen that the linearly approximated problem to the edge (1, 4) in G . Interestingly, this edge alone makes given by equations (24) and (25) can be solved efficiently if an edge dominating set. It is easy to see that the vertex the adjacency restriction is satisfied. Furthermore, it has corresponding to (1, 4) in L(G ) as illustrated in Figure 2 been proven theoretically that if each individual function in makes a dominating set. Similarly, (1, 2) in G (Figure 1(c)) the objective function is strictly convex and each individual and (1, 3) in G (Figure 1(c)) were chosen which are the function in constraints is convex for each relevant variable, elements in dominating sets of their line graphs. then the solution of the linearly approximated formulation In contrast to this, in Randic ´ index optimization, dif- without the adjacency restriction is feasible to the original ferent weights were allocated to different edges. %erefore, it problem [26], which, however, is not the case with our is natural to ask how Randic ´ index optimization deals with problem given by equations (10) and (11). Several numerical the graph symmetries. In particular, it is important to see if techniques are available in the literature to overcome this the edge equivalences are considered when assigning issue and to find approximate solutions [42, 48–50]. We weights. %erefore, we investigated the edge equivalences of adopted the scheme given by Markowitz and Manne [50] to these graphs to examine any possible connection to the generate our computational results. optimal weighted assignment. Edge equivalence of graphs is defined under global symmetry relations of graphs, char- acterized in terms of edge automorphisms. %is is defined in 4. Computational Results analogous to the notion of automorphisms in algebraic graph theory. %e automorphism group of a graph G is the We implemented the algorithms using the mixed-integer group formed by all structure-preserving permutations of its programming model in SageMath. %e experimentation vertices and is denoted by aut(G). Two vertices u and v in G took place over many graph structures up to 15 vertices. In are said to be structurally equivalent if there is an auto- particular, we tested all connected graphs up to 6 vertices. As morphism σ in aut(G) such that σ(u) � v. Edge automor- for Wiener index optimization, despite the triviality of the phism group of a graph G is defined as the automorphism formulation, some observations were of particular interest. group of the line graph L(G) of G defined as the graph For instance, the total weight was always assigned to a single obtained by associating a vertex with each edge of G and edge of the graph. Furthermore, this edge always belonged to connecting two vertices with an edge if the corresponding the edge dominating set of the graph (Table 1). %is is quite edges of G have a vertex in common. %us, the edge set of a natural, as the edges in the dominating set are the most graph can be classified into equivalence classes, in the sense significant in maintaining the connection strength of the of global symmetry. %e respective classification of the three graph, as each edge in the graph is either in the edge graphs in Figure 1 can be seen in Figure 3. dominating set or adjacent to at least one edge in the edge Now, the question can thus be restated as follows: if two dominating set. For instance, when the graph G in edges are structurally equivalent, does the optimal Randic ´ Figure 1(a) was considered for Wiener index optimization, index allocation assign equal weights to them? If yes, does it of which the results were derived making C in equation (3) 6 Advances in Operations Research Table 1: Optimal weight assignments for minimizing the Wiener index with edge dominating sets of graphs in Figure 1. Graph Nonzero weight assignment Edge dominating set G w(1, 4) � 1 (1, 4) { } G w(1, 2) � 1 {(1, 2), (1, 4)} G w(1, 3) � 1 (0, 4), (1, 3) { } 3 0 5 0 1 1 4 4 2 2 2 3 0 (a) (b) (c) Figure 1: %ree example graphs. (a) G . (b) G . (c) G . 1 2 3 (0, 1) (3, 4) (1, 3) (1, 4) (0, 4) (1, 2) (2, 4) Figure 2: %e line graph L(G ). 3 0 4 2 (a) (b) Figure 3: Continued. Advances in Operations Research 7 2 0 (c) Figure 3: Line graph of G and classification of the edge sets into equivalence classes for the graphs in Figure 1. (a) G . (b) G . (c) G . 1 1 2 3 always distinguish two nonequivalent pairs of edges? %e answers seemed to be negative. For instance, the edge au- tomorphism group of G is given by aut L G � ((1, 2), (1, 3))((2, 4), (3, 4)), ((0, 1), (0, 4))((1, 2), (2, 4))((1, 3), (3, 4)), ((0, 1), (1, 2))((0, 4), (2, 4)) , (26) 􏼁􏼁 { } which implies that (0, 1) and (0, 4) are structurally equiv- sense of dominating sets and global symmetry relations of a alent edges, as illustrated in Figure 3(a). However, according graph. to Table 2, they are allocated different weights. On the In a theoretical point of view, interesting comparisons contrary, same weight has been assigned to (1, 2) and (1, 4), can be made regarding optimizations of different topological despite they belong to a different equivalence class. %ere- indices. Recalling the optimization of the algebraic con- fore, it seems Randic index optimization does not reflect the nectivity (first nontrivial Laplacian eigenvalue) is convex, global symmetry of a graph. Having said that, it must be one can compare Wiener index optimization, of which the mentioned that, during our experimentation, we encoun- solution procedure is much simpler, and Randic ´ index tered many graphs for which the Randic ´ index optimization optimization, of which it is harder. %is may be extended was perfectly harmonious with edge classification by further by considering optimization of different topological equivalences. For instance, the three edge equivalence classes indices such as the Balaban index [51, 52], Harary index of G (Figure 3(b)) are distinguished by the optimal allo- [53, 54], Graovac–Pisanski index [55], and Hosoya index cation (Table 2), where each class is assigned its own weight [56]. On the contrary, it is interesting to investigate how unique to that class. these indices are related to the global symmetry of the graph. Although topological indices are studied extensively from 5. Discussion graph-theoretic perspectives, their relation to automor- phisms has not been paid much attention. It will be an Motivated by an epidemiological application and a previous interesting future work to consider if the optimal allocation work done by Ghosh et al. [8] related to electrical circuits, we of weights for other topological indices shows any relation considered the problem of assigning weights to edges of a with edge equivalences of graphs. It is noteworthy that a graph, subject to a total fixed edge weight, with the aim of recent work introduced a variant of the Wiener index, minimizing the connection strength of a graph, character- moderated in light of global symmetry of graphs as char- ized first by the Wiener index and then by the Randic index. acterized by the automorphism group [57]. It is natural to %ough these two topological indices are closely related to expect this version to resemble the global symmetry of the each other, in particular, a change in the Randic index results graph, of which the verification is left for future research in a change in the Wiener index, the two optimization studies. Furthermore, the relation of different topological problems of our consideration were far different from each indices to the edge dominating set could also be investigated. other. Wiener index optimization was trivial, as it was a In a practical point of view, different epidemiological linear formulation, while Randic ´ index optimization was a conditions might require optimization of different topo- nonlinear and a nonconvex optimization problem. %ere- logical indices. In the context of rapid transmission of a fore, we adopted the technique of separable programming to vector-borne disease, as our problem of interest, resources to find approximate solutions to Randic index optimization. control the disease must be assigned to edges Finally, we presented our computational experience in the 8 Advances in Operations Research Table 2: Optimal weight assignment for minimizing Randic ´ indices [5] I. Gutman and B. Mohar, “%e quasi-wiener and the kirchhoff of graphs in Figure 1. indices coincide,” Journal of Chemical Information and Computer Sciences, vol. 36, no. 5, pp. 982–985, 1996. Graph Weight [6] E. Otte and R. Rousseau, “Social network analysis: a powerful w(0, 1) � 0, w(0, 4) � 0.2, w(1, 2) � 0, w(1, 3) � 0.2, strategy, also for the information sciences,” Journal of In- w(1, 4) � 0, w(2, 4) � 0.6, w(3, 4) � 0 formation Science, vol. 28, no. 6, pp. 441–453, 2002. w(0, 1) � 0.4, w(1, 2) � 0, w(1, 3) � 0, w(1, 4) � 0, [7] M. Imran, S. Hayat, and M. Y. 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Optimizing Wiener and Randić Indices of Graphs

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Hindawi Advances in Operations Research Volume 2020, Article ID 3139867, 10 pages https://doi.org/10.1155/2020/3139867 Research Article Optimizing Wiener and Randic Indices of Graphs A. C. Mahasinghe , K. K. W. H. Erandi , and S. S. N. Perera Research & Development Center for Mathematical Modeling, Department of Mathematics, Faculty of Science, University of Colombo, Colombo 03, Sri Lanka Correspondence should be addressed to A. C. Mahasinghe; anuradhamahasinghe@gmail.com Received 9 June 2020; Accepted 4 August 2020; Published 26 September 2020 Academic Editor: Dylan F. Jones Copyright © 2020 A. C. Mahasinghe et al. %is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Wiener and Randi´ c indices have long been studied in chemical graph theory as connection strength measures of graphs. Later, these indices were used in different fields such as network analysis. We consider two optimization problems related to these indices, with potential applications to network theory, in particular to epidemiological networks. Given a connected graph and a fixed total edge weight, we investigate how individual weights must be assigned to edges, minimizing the connection strength of the graph. In order to measure the connection strength, we use the weighted Wiener index and a modified version of the ordinary ´ ´ Randic index. Wiener index optimization is linear, while Randic index optimization turns out to be both nonlinear and nonconvex. Hence, we adopt the technique of separable programming to generate solutions. We present our experimental results by applying relevant algorithms to several graphs. Mohar in 1996, in which it was proven that a variant of the 1. Introduction Wiener index coincides with the Kirchhoff index of electrical networks [5]. In 2002, Otte and Rousseau further extended Topological indices of graphs have served as numerical invariants of chemical structures, characterizing the to- the scope of topological indices by using them to analyze pology of the chemical structure graph theoretically. In social networks [6]. In a recent work, Imran et al. analyzed most cases, these indices were used to measure the interconnection networks using topological indices [7]. All connection strength of chemical compounds. %e first- these works used topological indices to characterize existing ever such topological index found in the literature was the networks with fixed vertex and edge weights in order to Wiener index, of which the intention was exploring derive information about the network. In contrast to this, thermodynamic and physiochemical properties of alkanes Ghosh et al. investigated how the edge weights can be in terms of molecular shapes [1].Consequently, variants of assigned subject to a fixed total weight, in order to optimize a the Wiener index and different other indices appeared for topological index, together with an application into electrical similar purposes, introducing a new field, chemical graph circuits [8]. %is application was interpreted as assigning resistors to the edges of an electric network, subject to a total theory, into theoretical chemistry [2–4]. %ough different indices intended for different characterizations of fixed sum of resistance, aimed at minimizing the total ef- chemical compounds, they shared in common the notion fective resistance or, analogously, maximizing the connec- of connection strength or compactness of the relevant tion strength. %e topological measure for the total effective graph structure. resistance has been taken as the algebraic connectivity %ough these indices were originally confined to the (smallest nontrivial Laplacian eigenvalue) of the relevant chemical graph theory, their scope has later been extended as graph, which is proven to be closely associated with the to include other subject areas as well. %e applicability of Wiener index [5]. Interestingly, the relevant optimization topological indices to networks beyond chemical structures problem turns out to be convex, guaranteeing efficient was initiated with the pioneering work by Gutman and solvability. 2 Advances in Operations Research the aim is the minimization of the compactness of the ep- Optimizing topological indices subject to different constraints in different contexts has been the subject of idemiological network, it is equivalent to optimizing an appropriate topological index subject to a fixed total edge interest in several previous works. Most of these optimi- zation problems were related to the chemical graph theory, weight. %is is the main problem of interest in this work. in particular to the design of chemical compounds. In [9], We first consider the Wiener index, which is the simplest Raman and Maranas developed an integer programming topological index for characterizing the compactness of the model to optimize a combination of topological indices network. However, in the context of epidemiological net- including Wiener and Randic ´ indices and Kier’s shape index works, a distance-based measure as Wiener index is less [10]. Optimizing the molecular interconnectivity index for significant than a degree-based measure as a region with many interconnections is likely to contribute significantly to the design of polymers had been discussed in [11], with a solution scheme for the resulting nonconvex mixed-integer the spread of the disease. We find the degree-based topo- logical index introduced by Randic ´ in 1975 [21] ideal for our linear programming formulation. A computational scheme for designing new molecules in medicinal chemistry was purpose. %ough this was originally intended for measuring the extent of branching of the carbon atom in hydrocarbons, described in [12] by Siddhaye et al., where the first-order molecular connectivity index was optimized through an similar to the Wiener index, later developments of the integer programming reformulation and the branch-and- Randic ´ index have proven its applicability in different bound approach. An optimization problem of a different contexts [22–24]. flavour in the context of chemical graph theory was con- Accordingly, we consider both Wiener index and Randic ´ sidered in [13], where the simplex algorithm was used to index to measure the compactness of the network in our derive optimal versions of several topological indices. optimization problem. %e problem of optimizing the Wiener index turns out to be linear, thus trivially solvable. A few works on optimizing topological indices appear outside the realm of chemical graph theory as well. A On the contrary, optimizing the Randic ´ index is a chal- lenging task as it turns out to be both nonlinear and non- concept paper by Preuß et al. [14] had proposed the opti- mization of Wiener and Randic ´ indices to solve the maxi- convex. In order to overcome the computational hardness, we adopt the technique of separable programming [25, 26] mum terrain coverage problem. A recent work [15] explored the possibility of optimizing the Wiener index for solving the and replace respective nonlinear functions by their piecewise critical node detection problem, where Benders algorithm linear approximations, eventually ending up with an ap- [16] was adopted as the solution technique. An application of proximate solution to the problem. algebraic connectivity maximization to communication %e remainder of the paper is organized as follows. In networks was discussed in [17]. Section 2, we reformulate the problem of minimizing the %e specific optimization problem considered by Ghosh Wiener index of a graph subject to a fixed total edge weight as a linear program. In Section 3, we consider the same et al. [8] is of particular interest and can be contrasted with the other optimization models with topological indices as optimization problem by replacing the Wiener index with the Randic ´ index, which turns out to be nonconvex. Our the optimization is done subject to a constant edge weight sum. It is natural to seek what the nature of the problem reformulation using separable programming techniques can would be if algebraic connectivity is replaced by a different be found in the same section. Section 4 contains our topological index. Would that be an efficiently solvable computational results, from which the discussion in Section optimization problem? Also, what if the objective was 5 is motivated. changed to minimizing the connection strength contrary to the electric network context? %ese are not merely questions 2. Optimizing the Wiener Index of theoretical interest; there is a useful application to epi- demiological networks. Consider the transmission of a %e simplest and the pioneering topological index of graphs vector-borne disease throughout a geographical region. %is is the Wiener index. Consider a simple connected undi- region can be considered as a network, of which the vertices rected graph G(V, E) with n vertices. %en, the Wiener index represent cities or suburbs, while the edges represent their W(G) of a graph G is defined as interconnections such as roads and channels, along which n n the vector-borne disease transmits [18]. %e weight of an W(G) � 􏽘 􏽘 dist(i, j), (1) edge in this network could be regarded as a measure of i�1 j�1 favorable conditions for breeding sites of vectors. It has been claimed that the rapid transmission of such a disease is where dist(i, j) is the distance between ith and jth vertices of largely influenced by the compactness of the network G. One may find several papers on finding Wiener indices of [19, 20]. %us, the health planners might be interested in different graphs [27–29]. %ough it is the unweighted ver- minimizing the compactness of this network by eliminating sion of the Wiener index which is used often, the vertex- the favorable conditions for vectors along the roads or water weighted graph version was introduced later [30], which has channels. However, this procedure is not without budgetary achieved progress in the past few years [31, 32]. Further- constraints. %e total amount of budget available in the more, an edge-weighted version (known as the Gutman control process for eliminating vector breeding sites must be index) was introduced as a natural extension of the Wiener optimally utilized along the roads and channels. %us, the index [33]. We follow this as the edge-weighted Wiener total edge weight must be bounded by a constant. Whenever index in our optimization problem. Accordingly, for a graph Advances in Operations Research 3 G together with functions d and w, where d is the degree of Aimed at minimizing the edge-weighted Randic index, the vertex i and w is the weight of the edge (i, j), the now we express the objective function to be minimized as ij weighted Wiener index is expressed as follows: Z � R (G), subject to the same budgetary and non- 2 w negativity constraints given by (3) and (4), respectively. %us, the problem turns out to be nonlinear. A closer W (G) � 􏽘 􏽘 w d d . (2) w ij i j i�1 look might reveal its nonconvexity. Recalling the problem of (i,j)∈E optimizing the topological index taken as algebraic con- Our objective is minimizing the function Z � W (G) in 1 w nectivity turned out to be convex, a closed-form solution was equation (2), subject to relevant constraints. In terms of the obtained [8]. In contrast to this, optimizing the Randic ´ index epidemiological network context, the sole major constraint is is a nonconvex optimization problem; thus, it is challenging the availability of budget for the control process. %is constraint to find the global optimum. Hence, instead of looking for enforces that the allocation of resources to roads and channels closed-form exact solutions, we seek an approximate solu- must be done subject to a fixed total budget. Let the total tion by using appropriate optimization and approximation (maximum possible) budget for resources be denoted by C. schemes. It is not difficult to see that our objective function %en, the relevant constraint can be expressed as in equation (6) can be easily convertible to a form expressible as sums of functions of individual decision variables. %is 􏽘 w ≤ C. motivates us to make use separable programming technique ij (3) (i,j)∈E for approximating the solution. Finally, the nonnegativity of w must be specified by ij 3.1. Separable Programming Formulation. %e method of ∀(i, j) ∈ E, w ≥ 0. (4) separable programming was first introduced for constrained ij optimization of nonlinear convex functions, whenever these It can be seen that the optimization problem given by functions are expressible as sums of functions of a single equations (2)–(4) is linear. %erefore, the solution is non- variable [25]. Functions with the latter mentioned property trivially obtained for Wiener index optimization. were called separable, and later works investigated the possibility of expanding the technique to nonconvex func- tions as well [42, 43]. Since its inception, separable pro- 3. Optimizing the Randic´ Index gramming has been a very useful optimization technique, with applications to several real problems including agri- %e topological index introduced for characterizing the con- cultural planning [44], linear complementarity problem nection strength of chemical compounds by Milan Randic ´ is [45], newsboy problem [46], and demand allocation [47]. widely known in chemical graph theory. %is has been a subject Notice that the objective function in equation (6) is of interest for graph theorists, and several works can be found nonlinear and nonconvex. Hence, we convert the objective on graph-theoretic aspects of the Randic ´ index [34–36]. Being function to a separable form. Let an elegant measure for the connection strength, this index has been applied in different contexts as well. Examples include measuring the robustness in cybernetics [22], reliability of a � 􏽘 w . (7) i ij communication networks [37], connectivity of mobile net- j�1 works [38], and information content of a graph [39]. %e ordinary Randic ´ index is expressed as follows: From equation (6), our objective function can be restated as 􏽰������� R(G) � 􏽘 . (5) √�� n 2 d(i)d(j) n n n (i,j)∈E 􏽐 a 􏼁 1 √��� � i�1 ⎝ ⎠ ⎛ ⎞ Z � � 􏽘 a + 􏽘 􏽘 a a . (8) 2 i i s C C i�1 i�1 s�1 Clearly, Randic ´ index is a degree-based topological in- dex, unlike the distance-based Wiener index. %us, unlike √�� √��� � Since a > 0 for all i ∈ {1, 2, . . . , n}, a a can be i i s Wiener index optimization which was more into theoretical replaced by y , where r ∈ {1, 2, . . . , n(n − 1)/2}. %en, it is interest, Randic ´ index optimization is directly relevant to our possible to transform the objective function to the separable epidemiological application mentioned in Section 1. form with the following constraint in the separable form: Moreover, a reduction of the Randic ´ index results in the √�� √�� reduction of the Wiener index, a fact which can be seen by log y � log a + log a . (9) 􏼁 􏼁 􏼁 r i s comparing equations (1) and (5) [40]. Now, the objective function can be restated as %e edge-weighted version of the Randic ´ index was introduced by Araujo and De la Peña as follows [41]: n ((n(n−1))/2) ⎛ ⎝ ⎞ ⎠ 􏽱�������� � Z � a + 2 􏽘 y . (10) 2 􏽘 2 i r 􏽐 􏽐 w 􏼒 􏼓 i�1 r�1 i�1 (i,j)∈E ij (6) R (G) � . 􏽐 􏽐 w %is is to be minimized subject to i�1 (i,j)∈E ij 4 Advances in Operations Research 􏽘 􏽘 w ≤ C, (11a) ij i�1 j∈V a � 􏽘 w , ∀i ∈ 1, 2, . . . , n , { } (11b) i ij j�1 1, 2, . . . , n(n − 1) √� � √�� log y 􏼁 � log a 􏼁 + log a 􏼁 , ∀r ∈ 􏼨 􏼩, (11c) r i s w ≥ 0, ∀i, j ∈ {1, 2, . . . , n}, (11d) ij a > 0, ∀i ∈ {1, 2, . . . , n}, (11e) 1, 2, . . . , n(n − 1) y > 0, ∀r ∈ 􏼨 􏼩. (11f ) 3.2. Linearly Approximated Program. Notice that the con- straint given by equation (11c) is nonlinear. In order to y � 􏽘 λ α , r rk rk k�1 approximate by piecewise linear functions, first, we restate (19) equation (11c) as 􏽘 λ � 1. 2 rk 1, 2, . . . , n(n − 1) k�1 g y 􏼁 + 􏽘 g 􏼐x 􏼑 � 0, ∀r ∈ 􏼨 􏼩, r1 r r(1+p) p p�1 Let the domain of g (x ) be the interval [C , C]. %en, rp p 1 (12) we divide the interval into h subdivisions of length q by defining β as follows: pl where g y 􏼁 � log y , (13a) C � β ≤ β ≤ · · · ≤ β � C, (20) r1 r r 0 p1 p2 ph √�� where g x 􏼁 � −log a , (13b) r2 1 i 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 √�� 􏼌 􏼌 β − β � q. (21) 􏼌 􏼌 pl p(l+1) g x 􏼁 � −log a . (13c) r3 2 s %en, the piecewise linear approximation to g (x ) can Let the domain of g (y ) be the interval [C , C]. %en, rp p r1 r 0 be expressed as we divide the interval into m subdivisions of length d by defining α as follows: rk (22) C � α ≤ α ≤ · · · ≤ α � C, (14) g 􏼐x 􏼑 � 􏽘 μ β , 0 r1 r2 rm rp p pl pl l�1 where where 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 α − α � d. (15) 􏼌 􏼌 rk r(k+1) x � 􏽘 μ β , p pl pl Now, a point y ∈ [C , C] can be uniquely expressed as r 0 l�1 (23) y � λ α + λ α � 1, (16) r rk rk r(k+1) r(k+1) h 􏽘 μ � 1. pl where l�1 λ + λ � 1. (17) rk r(k+1) Now, the nonlinear program is approximated by the following problem: %en, the piecewise linear approximation to g (y ) can r1 r ((n(n−1))/2) be expressed as m n h ⎝ ⎠ ⎛ ⎞ minimize Z � 􏽘 􏽘 λ α + 􏽘 􏽘 μ β , m 3 rk rk il il r�1 i�1 k�1 l�1 g y 􏼁 � 􏽘 λ g α 􏼁 , (18) r1 r rk r1 rk k�1 (24) where subject to Advances in Operations Research 5 􏽘 􏽘 w ≤ C, (25a) ij i�1 j∈V n h 􏽘 􏽘 μ β � 􏽘 w , (25b) il il ij i�1 l�1 (i,j)∈E m 2 h 1, 2, . . . , n(n − 1) 􏽘 λ g α 􏼁 + 􏽘 􏽘 μ g 􏼐β 􏼑 � 0, ∀r ∈ 􏼨 􏼩, (25c) rk r1 rk pl rp pl p�1 k�1 l�1 1, 2, . . . , n(n − 1) 􏽘 λ � 1, ∀r ∈ 􏼨 􏼩, (25d) rk k�1 1, 2, . . . , n(n − 1) λ , α ≥ 0, ∀r ∈ 􏼨 􏼩, ∀k ∈ {1, 2, . . . , m,} (25e) rk rk 􏽘 μ � 1, ∀i ∈ {1, 2, . . . , n}, (25f ) il l�1 μ , β ≥ 0, ∀i ∈ {1, 2, . . . , n}, ∀l ∈ {1, 2, . . . , h}, (25g) il il and at most two adjacent λ ’s are positive. equal to one, the optimal solution assigned a total weight of 1 rk It can be seen that the linearly approximated problem to the edge (1, 4) in G . Interestingly, this edge alone makes given by equations (24) and (25) can be solved efficiently if an edge dominating set. It is easy to see that the vertex the adjacency restriction is satisfied. Furthermore, it has corresponding to (1, 4) in L(G ) as illustrated in Figure 2 been proven theoretically that if each individual function in makes a dominating set. Similarly, (1, 2) in G (Figure 1(c)) the objective function is strictly convex and each individual and (1, 3) in G (Figure 1(c)) were chosen which are the function in constraints is convex for each relevant variable, elements in dominating sets of their line graphs. then the solution of the linearly approximated formulation In contrast to this, in Randic ´ index optimization, dif- without the adjacency restriction is feasible to the original ferent weights were allocated to different edges. %erefore, it problem [26], which, however, is not the case with our is natural to ask how Randic ´ index optimization deals with problem given by equations (10) and (11). Several numerical the graph symmetries. In particular, it is important to see if techniques are available in the literature to overcome this the edge equivalences are considered when assigning issue and to find approximate solutions [42, 48–50]. We weights. %erefore, we investigated the edge equivalences of adopted the scheme given by Markowitz and Manne [50] to these graphs to examine any possible connection to the generate our computational results. optimal weighted assignment. Edge equivalence of graphs is defined under global symmetry relations of graphs, char- acterized in terms of edge automorphisms. %is is defined in 4. Computational Results analogous to the notion of automorphisms in algebraic graph theory. %e automorphism group of a graph G is the We implemented the algorithms using the mixed-integer group formed by all structure-preserving permutations of its programming model in SageMath. %e experimentation vertices and is denoted by aut(G). Two vertices u and v in G took place over many graph structures up to 15 vertices. In are said to be structurally equivalent if there is an auto- particular, we tested all connected graphs up to 6 vertices. As morphism σ in aut(G) such that σ(u) � v. Edge automor- for Wiener index optimization, despite the triviality of the phism group of a graph G is defined as the automorphism formulation, some observations were of particular interest. group of the line graph L(G) of G defined as the graph For instance, the total weight was always assigned to a single obtained by associating a vertex with each edge of G and edge of the graph. Furthermore, this edge always belonged to connecting two vertices with an edge if the corresponding the edge dominating set of the graph (Table 1). %is is quite edges of G have a vertex in common. %us, the edge set of a natural, as the edges in the dominating set are the most graph can be classified into equivalence classes, in the sense significant in maintaining the connection strength of the of global symmetry. %e respective classification of the three graph, as each edge in the graph is either in the edge graphs in Figure 1 can be seen in Figure 3. dominating set or adjacent to at least one edge in the edge Now, the question can thus be restated as follows: if two dominating set. For instance, when the graph G in edges are structurally equivalent, does the optimal Randic ´ Figure 1(a) was considered for Wiener index optimization, index allocation assign equal weights to them? If yes, does it of which the results were derived making C in equation (3) 6 Advances in Operations Research Table 1: Optimal weight assignments for minimizing the Wiener index with edge dominating sets of graphs in Figure 1. Graph Nonzero weight assignment Edge dominating set G w(1, 4) � 1 (1, 4) { } G w(1, 2) � 1 {(1, 2), (1, 4)} G w(1, 3) � 1 (0, 4), (1, 3) { } 3 0 5 0 1 1 4 4 2 2 2 3 0 (a) (b) (c) Figure 1: %ree example graphs. (a) G . (b) G . (c) G . 1 2 3 (0, 1) (3, 4) (1, 3) (1, 4) (0, 4) (1, 2) (2, 4) Figure 2: %e line graph L(G ). 3 0 4 2 (a) (b) Figure 3: Continued. Advances in Operations Research 7 2 0 (c) Figure 3: Line graph of G and classification of the edge sets into equivalence classes for the graphs in Figure 1. (a) G . (b) G . (c) G . 1 1 2 3 always distinguish two nonequivalent pairs of edges? %e answers seemed to be negative. For instance, the edge au- tomorphism group of G is given by aut L G � ((1, 2), (1, 3))((2, 4), (3, 4)), ((0, 1), (0, 4))((1, 2), (2, 4))((1, 3), (3, 4)), ((0, 1), (1, 2))((0, 4), (2, 4)) , (26) 􏼁􏼁 { } which implies that (0, 1) and (0, 4) are structurally equiv- sense of dominating sets and global symmetry relations of a alent edges, as illustrated in Figure 3(a). However, according graph. to Table 2, they are allocated different weights. On the In a theoretical point of view, interesting comparisons contrary, same weight has been assigned to (1, 2) and (1, 4), can be made regarding optimizations of different topological despite they belong to a different equivalence class. %ere- indices. Recalling the optimization of the algebraic con- fore, it seems Randic index optimization does not reflect the nectivity (first nontrivial Laplacian eigenvalue) is convex, global symmetry of a graph. Having said that, it must be one can compare Wiener index optimization, of which the mentioned that, during our experimentation, we encoun- solution procedure is much simpler, and Randic ´ index tered many graphs for which the Randic ´ index optimization optimization, of which it is harder. %is may be extended was perfectly harmonious with edge classification by further by considering optimization of different topological equivalences. For instance, the three edge equivalence classes indices such as the Balaban index [51, 52], Harary index of G (Figure 3(b)) are distinguished by the optimal allo- [53, 54], Graovac–Pisanski index [55], and Hosoya index cation (Table 2), where each class is assigned its own weight [56]. On the contrary, it is interesting to investigate how unique to that class. these indices are related to the global symmetry of the graph. Although topological indices are studied extensively from 5. Discussion graph-theoretic perspectives, their relation to automor- phisms has not been paid much attention. It will be an Motivated by an epidemiological application and a previous interesting future work to consider if the optimal allocation work done by Ghosh et al. [8] related to electrical circuits, we of weights for other topological indices shows any relation considered the problem of assigning weights to edges of a with edge equivalences of graphs. It is noteworthy that a graph, subject to a total fixed edge weight, with the aim of recent work introduced a variant of the Wiener index, minimizing the connection strength of a graph, character- moderated in light of global symmetry of graphs as char- ized first by the Wiener index and then by the Randic index. acterized by the automorphism group [57]. It is natural to %ough these two topological indices are closely related to expect this version to resemble the global symmetry of the each other, in particular, a change in the Randic index results graph, of which the verification is left for future research in a change in the Wiener index, the two optimization studies. Furthermore, the relation of different topological problems of our consideration were far different from each indices to the edge dominating set could also be investigated. other. Wiener index optimization was trivial, as it was a In a practical point of view, different epidemiological linear formulation, while Randic ´ index optimization was a conditions might require optimization of different topo- nonlinear and a nonconvex optimization problem. %ere- logical indices. In the context of rapid transmission of a fore, we adopted the technique of separable programming to vector-borne disease, as our problem of interest, resources to find approximate solutions to Randic index optimization. control the disease must be assigned to edges Finally, we presented our computational experience in the 8 Advances in Operations Research Table 2: Optimal weight assignment for minimizing Randic ´ indices [5] I. Gutman and B. Mohar, “%e quasi-wiener and the kirchhoff of graphs in Figure 1. indices coincide,” Journal of Chemical Information and Computer Sciences, vol. 36, no. 5, pp. 982–985, 1996. Graph Weight [6] E. Otte and R. Rousseau, “Social network analysis: a powerful w(0, 1) � 0, w(0, 4) � 0.2, w(1, 2) � 0, w(1, 3) � 0.2, strategy, also for the information sciences,” Journal of In- w(1, 4) � 0, w(2, 4) � 0.6, w(3, 4) � 0 formation Science, vol. 28, no. 6, pp. 441–453, 2002. w(0, 1) � 0.4, w(1, 2) � 0, w(1, 3) � 0, w(1, 4) � 0, [7] M. Imran, S. Hayat, and M. Y. 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