A Cheeger type inequality in finite Cayley sum graphs

Abstract

Let G be a finite group with |G| > 4 and S be a subset of G with |S| = d such that the Cayley sum graph CΣ(G, S) is undirected and connected. We show that the nontrivial spectrum of the normalised adjacency operator of CΣ(G, S) is controlled by its Cheeger constant and its degree. We establish an explicit lower bound for the non-trivial spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval ( − 1 + hΣ(G)4 , 1 − hΣ(G)2 ), where hΣ(G) denotes the vertex Cheeger constant η 2d2 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 Cayley graph of finite groups.