Symmetries of finite graphs and homology

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 ¯$\overline{\Gamma }$ .