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 distance-regular graphs with diameter 3, and most distance-regular graphs with small valency.

中文翻译：

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距离正则图的Cheeger常数

图的Cheeger常数是子图的大小与其边界的大小之间的最小可能比率。众所周知，此常数必须至少为$\frac{{\lambda }_{\mathrm{1个}}}{2}$，在哪里 ${\lambda }_{\mathrm{1个}}$是拉普拉斯矩阵的最小正特征值。本文的主题是作者的一个猜想，即对于距离正则图，Cheeger常数最大为${\lambda }_{\mathrm{1个}}$。特别是，我们证明了已知的无穷远距离规则图，直径为2的距离规则图（强正则图），几类直径为3的距离规则图以及大多数具有小价。