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

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

1998-06-08

arxiv.org/abs/math/9806037v1

Mathematics

Combinatorics

AMSTeX, to appear in Trans. Amer. Math. Soc

Scientific paper

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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