License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSTTCS.2017.46
URN: urn:nbn:de:0030-drops-83861
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8386/
Rangarajan, Bharatram
A Combinatorial Proof of Ihara-Bass's Formula for the Zeta Function of Regular Graphs
Abstract
We give an elementary combinatorial proof of Bass's determinant formula for the zeta function of a finite regular graph. This is done by expressing the number of non-backtracking cycles of a given length in terms of Chebyshev polynomials in the eigenvalues of the adjacency operator of the graph. A related observation of independent interest is that the Ramanujan property of a regular graph is equivalent to tight bounds on the number of non-backtracking cycles of every length.
BibTeX - Entry
@InProceedings{rangarajan:LIPIcs:2018:8386,
author = {Bharatram Rangarajan},
title = {{A Combinatorial Proof of Ihara-Bass's Formula for the Zeta Function of Regular Graphs}},
booktitle = {37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)},
pages = {46:46--46:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-055-2},
ISSN = {1868-8969},
year = {2018},
volume = {93},
editor = {Satya Lokam and R. Ramanujam},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/8386},
URN = {urn:nbn:de:0030-drops-83861},
doi = {10.4230/LIPIcs.FSTTCS.2017.46},
annote = {Keywords: non-backtracking, Ihara zeta, Chebyshev polynomial, Ramanujan graph, Hashimoto matrix}
}
Keywords: |
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non-backtracking, Ihara zeta, Chebyshev polynomial, Ramanujan graph, Hashimoto matrix |
Collection: |
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37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017) |
Issue Date: |
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2018 |
Date of publication: |
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12.02.2018 |