  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 $\frac{\lambda_1}{2}$, where $\lambda_1$ 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 $\lambda_1$. 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 imprimitive
distance-regular graphs, and most distance-regular graphs with small valency.
