On cliques in edge-regular graphs

Abstract

Let Γ be an edge-regular graph with given parameters (v,k,λ). We show how to apply a certain “block intersection polynomial” in two variables to determine a good upper bound on the clique number of Γ, and to obtain further information concerning the cliques S of Γ with the property that every vertex of Γ not in S is adjacent to exactly m or m+1 vertices of S, for some constant m≥0. Some interesting examples are studied using computation with groups and graphs.