A Combinatorial Proof of Bass's Evaluations of the Ihara-Selberg Zeta Function for Graphs
Combinatorics
2007-05-23 v1
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.
Keywords
Cite
@article{arxiv.math/9806037,
title = {A Combinatorial Proof of Bass's Evaluations of the Ihara-Selberg Zeta Function for Graphs},
author = {Dominique Foata and Doron Zeilberger},
journal= {arXiv preprint arXiv:math/9806037},
year = {2007}
}
Comments
AMSTeX, to appear in Trans. Amer. Math. Soc