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On the local geometry of graphs in terms of their spectra

Huang, Brice and Rahman, Mustazee (2019) 'On the local geometry of graphs in terms of their spectra.', European journal of combinatorics., 81 . pp. 378-393.

Abstract

In this paper we consider the relation between the spectrum and the number of short cycles in large graphs. Suppose G1,G2,G3,... is a sequence of finite and connected graphs that share a common universal cover T and such that the proportion of eigenvalues of Gn that lie within the support of the spectrum of T tends to 1 in the large 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.

Item Type:Article
Full text:(AM) Accepted Manuscript
Available under License - Creative Commons Attribution Non-commercial No Derivatives 4.0.
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1016/j.ejc.2019.07.001
Publisher statement:© 2021 This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Date accepted:03 July 2019
Date deposited:06 October 2021
Date of first online publication:10 July 2019
Date first made open access:06 October 2021

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