Groups and Graphs

Abstract

Let G be a finite group, H a subgroup, and S a subset of G. We can define a graph Г (G,H,S) by taking as vertices the cosets gH (g ∈ G) and calling g1H and g2H adjacent when $Hg_2^{-1} g1H \subseteq HSH$. The group G acts as a group of automorphisms on Г(G,H,S) via left multiplication, and this action is transitive. The stabilizer of the vertex H is the subgroup H. A graph Γ (G,H,S) with H = 1 is called a Cayley graph.