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Symmetries of finite graphs and homology

Symmetries of finite graphs and homology 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 ¯ $\overline{\Gamma }$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups Complexity Cryptology de Gruyter

Symmetries of finite graphs and homology

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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

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 ¯ $\overline{\Gamma }$ .

Journal

Groups Complexity Cryptologyde Gruyter

Published: May 1, 2015

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