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arXiv · 1811.00230

On the Cheeger constant for distance-regular graphs

Jack Koolen·Greg Markowsky·Zhi Qiao
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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 $\frac{\lambda_1}{2}$, 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.

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Jack Koolen, Greg Markowsky, Zhi Qiao. 2018-11-01. On the Cheeger constant for distance-regular graphs. https://arxiv.org/abs/1811.00230

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