Graph eigenvalues play a fundamental role in controlling structural properties which are critical considerations in the design of supercomputing interconnection networks, such as bisection bandwidth, diameter, and fault tolerance. This motivates considering graphs with optimal spectral expansion, called Ramanujan graphs , as potential candidates for interconnection networks. In this work, we explore this possibility by comparing Ramanujan graph properties against those of a wide swath of current and proposed supercomputing topologies. We derive analytic expressions for the spectral gap, bisection bandwidth, and diameter of these topologies, some of which were previously unknown. We find the spectral gap of existing topologies is well separated from the optimal achievable by Ramanujan topologies, suggesting the potential utility of adopting Ramanujan graphs as interconnection networks.

中文翻译：

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拉马努金图和超级计算拓扑的光谱差距

图特征值在控制结构特性方面发挥着重要作用，这些特性是超级计算互连网络设计中的关键考虑因素，例如二分带宽、直径和容错性。这促使考虑具有最佳频谱扩展的图，称为拉马努金图，作为互连网络的潜在候选者。在这项工作中，我们通过将拉马努金图属性与大量当前和提议的超级计算拓扑的属性进行比较来探索这种可能性。我们推导出这些拓扑的光谱间隙、二等分带宽和直径的解析表达式，其中一些以前是未知的。我们发现现有拓扑的光谱差距与拉马努金拓扑可实现的最佳分离，