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. Download PDF (424Kb) |
| 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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