We revisit the classical question of the relationship between the diameter of a graph and its expansion properties. One direction is well understood: expander graphs exhibit essentially the lowest possible diameter. We focus on the reverse direction, showing that “sufficiently large” graphs of fixed diameter and degree must be “good” expanders. We prove this statement for various definitions of “sufficiently large” (multiplicative/additive factor from the largest possible size), for different forms of expansion (edge, vertex, and spectral expansion), and for both directed and undirected graphs. A recurring theme is that the lower the diameter of the graph and (more importantly) the larger its size, the better the expansion guarantees. Aside from inherent theoretical interest, our motivation stems from the domain of network design. Both low-diameter networks and expanders are prominent approaches to designing high-performance networks in parallel computing, HPC, datacenter networking, and beyond. Our results establish that these two approaches are, in fact, inextricably intertwined. We leave the reader with many intriguing questions for future research.

我们重新讨论图的直径与其扩展特性之间关系的经典问题。一个方向已广为人知：膨胀图基本上显示了最小的直径。我们专注于相反的方向，表明固定直径和度数的“足够大”图必须是“良好”的扩展器。我们针对“足够大”（各种可能的最大尺寸的乘法/加法因子）的各种定义，不同形式的扩展（边缘，顶点和频谱扩展）以及有向图和无向图证明了该说法。一个经常出现的主题是，图的直径越小，（更重要的是）图的大小越大，扩展保证就越好。除了固有的理论兴趣外，我们的动机还来自网络设计领域。小直径网络和扩展器都是在并行计算，HPC，数据中心网络等领域设计高性能网络的杰出方法。我们的结果表明，这两种方法实际上是密不可分的。我们给读者留下了许多有趣的问题，可供将来研究。