The r-th power of a graph modifies a graph by connecting every vertex pair within distance r. This paper gives a generalization of the Alon-Boppana Theorem for the r-th power of graphs, including irregular graphs. This leads to a generalized notion of Ramanujan graphs, those for which the powered graph has a spectral gap matching the derived Alon-Boppana bound. In particular, we show that certain graphs that are not good expanders due to local irregularities, such as Erdos-Renyi random graphs, become almost Ramanujan once powered. A different generalization of Ramanujan graphs can also be obtained from the nonbacktracking operator. We next argue that the powering operator gives a more robust notion than the latter: Sparse Erdos-Renyi random graphs with an adversary modifying a subgraph of log(n)^c$ vertices are still almost Ramanujan in the powered sense, but not in the nonbacktracking sense. As an application, this gives robust community testing for different block models.

中文翻译：

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幂图和广义拉马努金图的 Alon-Boppana 定理

图的 r 次幂通过连接距离 r 内的每个顶点对来修改图。本文给出了 Alon-Boppana 定理对图的 r 次幂的推广，包括不规则图。这导致了拉马努金图的广义概念，其中幂图具有与导出的 Alon-Boppana 界限匹配的光谱间隙。特别是，我们表明某些由于局部不规则而不是很好的扩展器的图，例如 Erdos-Renyi 随机图，一旦上电就几乎变成了 Ramanujan。还可以从非回溯算子中获得拉马努金图的不同概括。我们接下来认为，powering operator 给出了比后者更强大的概念：带有修改 log(n)^c$ 顶点子图的对手的稀疏 Erdos-Renyi 随机图在幂意义上仍然几乎是拉马努金，但在非回溯意义上则不是。作为一个应用程序，这为不同的块模型提供了强大的社区测试。