On the Cheeger constant for distance-regular graphs

Abstract

The Cheeger constant of a graph is the smallest possible ratio between the size of a subgraph and the size of its boundary. It is well known that this constant must be at least λ12, where λ1 is the smallest positive eigenvalue of the Laplacian matrix. The subject of this paper is a conjecture of the authors that for distance-regular graphs the Cheeger constant is at most λ1. In particular, we prove the conjecture for the known infinite families of distance-regular graphs, distance-regular graphs of diameter 2 (the strongly regular graphs), several classes of distance-regular graphs with diameter 3, and most distance-regular graphs with small valency.