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Groups Complexity Cryptology
, Volume 7 (1) – May 1, 2015

/lp/de-gruyter/symmetries-of-finite-graphs-and-homology-j2lGdPeZZ1

- Publisher
- de Gruyter
- Copyright
- Copyright © 2015 by the
- ISSN
- 1867-1144
- eISSN
- 1869-6104
- DOI
- 10.1515/gcc-2015-0003
- Publisher site
- See Article on Publisher Site

Abstract A finite symmetric graph Γ is a pair ( Γ , f ) $(\Gamma ,f)$ , where Γ is a finite graph and f : Γ → Γ $f:\Gamma \rightarrow \Gamma $ is a graph self equivalence or automorphism. We develop several tools for studying such symmetries. In particular, we describe in detail all symmetries with a single edge orbit, we prove that each symmetric graph has a maximal forest that meets each edge orbit in a sequential set of edges – a sequential maximal forest – and we calculate the characteristic polynomial χ f ( t ) $\chi _f(t)$ and the minimal polynomial μ f ( t ) $\mu _f(t)$ of the linear map H 1 ( f ) : H 1 ( Γ , ℤ ) → H 1 ( Γ , ℤ ) $H_1(f):H_1(\Gamma ,\mathbb {Z})\rightarrow H_1(\Gamma ,\mathbb {Z})$ . The calculation is in terms of the quotient graph Γ ¯ $\overline{\Gamma }$ .

Groups Complexity Cryptology – de Gruyter

**Published: ** May 1, 2015

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