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Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Ciobanu, Laura / Conder, Marston / Eick, Bettina / Elder, Murray / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura / Lohrey, Markus

CiteScore 2018: 0.80

SCImago Journal Rank (SJR) 2018: 0.368
Source Normalized Impact per Paper (SNIP) 2018: 1.061

Mathematical Citation Quotient (MCQ) 2018: 0.38

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Symmetries of finite graphs and homology

Benjamin Atchison / Edward C. Turner
Published Online: 2015-04-14 | DOI: https://doi.org/10.1515/gcc-2015-0003


A finite symmetric graph Γ is a pair (Γ,f), where Γ is a finite graph and f:ΓΓ 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,)H1(Γ,). 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

About the article

Received: 2014-05-19

Published Online: 2015-04-14

Published in Print: 2015-05-01

Funding Source: Framingham State University

Award identifier / Grant number: CELTSS

Citation Information: Groups Complexity Cryptology, Volume 7, Issue 1, Pages 11–30, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2015-0003.

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