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arXiv · 2006.11248

An Alon-Boppana theorem for powered graphs and generalized Ramanujan graphs

Emmanuel Abbe·Peter Ralli
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Abstract

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.

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Emmanuel Abbe, Peter Ralli. 2020-06-18. An Alon-Boppana theorem for powered graphs and generalized Ramanujan graphs. https://arxiv.org/abs/2006.11248

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