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.
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.
|Full text:||(AM) Accepted Manuscript|
Available under License - Creative Commons Attribution Non-commercial No Derivatives 4.0.
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|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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