A Cheeger type inequality in finite Cayley sum graphs

Arindam Biswas,Jyoti Prakash Saha

doi:10.5802/alco.166

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 C Σ (G,S) is undirected and connected. We show that the non-trivial spectrum of the normalised adjacency operator of C Σ (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 + h Σ (G) 4 η , 1 - h Σ (G) 2 2d 2 , where h Σ (G) denotes the vertex Cheeger constant of the d-regular graph C Σ (G,S) and η=2 9 d 8 . 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.

Highlights

Let G be a finite group, and S be a subset of G with |S| = d
The Cayley sum graph CΣ(G, S) is the graph having G as its set of vertices and for g, h ∈ G, the vertex h is adjacent to g if h = g−1s for some element s ∈ S
We recall that the Cayley graph of G, denoted by C(G, S), is the graph having G as its set of vertices and a vertex h is adjacent to a vertex g if h = gs for some element s ∈ S

Summary

Introduction

Let hΣ(G) denote the vertex Cheeger constant of the connected Cayley sum graph CΣ(G, S). Theorem 1.4 focuses on the non-bipartite Cayley sum graphs and shows that the smallest eigenvalue of the normalised adjacency matrix admits a lower bound depending only on the vertex Cheeger constant and the degree. We assume on the contrary that the normalised adjacency matrix T of the Cayley sum graph admits an eigenvalue close to −1 (see Theorem 2.10).

Results

Conclusion