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 $\frac{\lambda_1}{2}$, 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.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2018
- DOI:
- 10.48550/arXiv.1811.00230
- arXiv:
- arXiv:1811.00230
- Bibcode:
- 2018arXiv181100230K
- Keywords:
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- Mathematics - Combinatorics