Spectra of twists of Cayley and Cayley sum graphs

Abstract

Let G be a finite group with |G|≥4 and S be a subset of G. Given an automorphism σ of G, the twisted Cayley graph C(G,S)σ is the graph with G as its set of vertices, and the neighbourhood of a vertex is obtained by applying σ to its neighbourhood in the Cayley graph of G with respect to S. If C(G,S)σ is undirected and connected, then we prove that the nontrivial spectrum of its normalised adjacency operator is bounded away from −1 and this bound depends only on its degree, the order of σ and the vertex Cheeger constant of C(G,S)σ. The twisted Cayley sum graph CΣ(G,S)σ is defined similarly and we establish an analogous result for it. Further, we prove an analogous result for the Schreier graphs satisfying certain conditions.