A Combinatorial Proof of Bass’s Evaluations of the Ihara-Selberg Zeta Function for Graphs
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- by Dominique Foata and Doron Zeilberger PDF
- Trans. Amer. Math. Soc. 351 (1999), 2257-2274 Request permission
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.References
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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
- 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.
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 2257-2274
- MSC (1991): Primary 05C05, 05C25, 05C50; Secondary 11F72, 15A15, 16A27
- DOI: https://doi.org/10.1090/S0002-9947-99-02234-5
- MathSciNet review: 1487614
Dedicated: This paper is dedicated to Gian-Carlo Rota, on his millionth$_{2}$’s birthday.