A Cheeger Type Inequality in Finite Cayley Sum Graphs

Abstract

Let G be a finite group and S be a symmetric generating set of G with |S|=d. We show that if the undirected Cayley sum graph CΣ(G,S) is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from −1. We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval (−1+h(G)4η,1−h(G)22d2], where h(G) denotes the (vertex) Cheeger constant of the d-regular graph CΣ(G,S) and η=29d8. Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the non-bipartite Cayley graph C(G,S).