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arXiv · 1807.06034

On the Local Geometry of Graphs in Terms of Their Spectra

Brice Huang·Mustazee Rahman
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Abstract

In this paper we consider the relation between the spectrum and the number of short cycles in large graphs. Suppose $G_1, G_2, G_3, \ldots$ is a sequence of finite and connected graphs that share a common universal cover $T$ and such that the proportion of eigenvalues of $G_n$ that lie within the support of the spectrum of $T$ tends to 1 in the large $n$ limit. This is a weak notion of being Ramanujan. We prove such a sequence of graphs is asymptotically locally tree-like. This is deduced by way of an analogous theorem proved for certain infinite sofic graphs and unimodular networks, which extends results for regular graphs and certain infinite Cayley graphs.

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BibTeXRIS

Brice Huang, Mustazee Rahman. 2018-07-16. On the Local Geometry of Graphs in Terms of Their Spectra. https://doi.org/10.1016/j.ejc.2019.07.001

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