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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A Combinatorial Proof of Bass's Evaluations
of the Ihara-Selberg Zeta Function for Graphs


Authors: Dominique Foata and Doron Zeilberger
Journal: Trans. Amer. Math. Soc. 351 (1999), 2257-2274
MSC (1991): Primary 05C05, 05C25, 05C50; Secondary 11F72, 15A15, 16A27
Published electronically: February 8, 1999
MathSciNet review: 1487614
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Abstract | References | Similar Articles | Additional Information

Abstract: We derive combinatorial proofs of the main two evaluations of the Ihara-Selberg zeta function associated with a graph. We give three proofs of the first evaluation all based on the algebra of Lyndon words. In the third proof it is shown that the first evaluation is an immediate consequence of Amitsur's identity on the characteristic polynomial of a sum of matrices. The second evaluation of the Ihara-Selberg zeta function is first derived by means of a sign-changing involution technique. Our second approach makes use of a short matrix-algebra argument.


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Additional Information

Dominique Foata
Affiliation: Département de Mathématique, Université Louis Pasteur, 7, rue René-Descartes, F-67084 Strasbourg, France
Email: foata@math.u-strasbg.fr

Doron Zeilberger
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
Email: zeilberg@math.temple.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02234-5
PII: S 0002-9947(99)02234-5
Keywords: Ihara-Selberg zeta function, Lyndon words, Amitsur identity
Received by editor(s): March 2, 1997
Published electronically: February 8, 1999
Additional Notes: The second author was supported in part by N.S.F. and the first author as a consultant of Zeilberger on his grant.
Dedicated: This paper is dedicated to Gian-Carlo Rota, on his millionth$_{2}$’s birthday.
Article copyright: © Copyright 1999 American Mathematical Society