Symmetries of finite graphs and homology

Benjamin Atchison; Edward C. Turner

doi:10.1515/gcc-2015-0003

Published by De Gruyter April 14, 2015

Symmetries of finite graphs and homology

Benjamin Atchison and Edward C. Turner

From the journal Groups Complexity Cryptology

https://doi.org/10.1515/gcc-2015-0003

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Abstract

A finite symmetric graph Γ is a pair (Γ,f), where Γ is a finite graph and f:Γ→Γ 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 χf(t) and the minimal polynomial μf(t) of the linear map H1(f):H1(Γ,ℤ)→H1(Γ,ℤ). The calculation is in terms of the quotient graph Γ¯.

Keywords: Characteristic polynomial; minimal polynomial; finite graph; homology; sequential maximal forest; symmetry

MSC: 20E05; 20F28; 05C20; 05C25; 05C50

Funding source: Framingham State University

Award Identifier / Grant number: CELTSS

We are especially thankful for the encouragement of our family, friends, colleagues and respective departments.

Received: 2014-5-19

Published Online: 2015-4-14

Published in Print: 2015-5-1

Symmetries of finite graphs and homology

Abstract

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