``Combinatorial Proofs of Bass's Evaluation of the Ihara-Selberg Zeta function for a Graph''
By Dominique Foata and Doron Zeilberger
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(Appeared in Trans. Amer. Math. Soc. 351 (1999), 2257--2274. )

Written: Dec. 1, 1996.

Combinatorics has come a long way since H.C. Whitehead (1904-1960) had called graph theory `the slum of topology' and Jean Dieudonné had disdainfully enclosed combinatorics in quotation marks stating that ``many problems arising in `combinatorics' are without issue.'' (A Panorama of Pure Mathematics, Academic Press, 1982, p.2).

What could be a better proof that the tables have turned than the fact that the two persons that come right after Whitehead in Dieudonné's list of the originators of K-theory (ibid, p. 180), Jean-Pierre Serre and Hyman Bass, don't hesitate to use trees and graph theory in their own works.

In this paper, Dominique Foata and I give purely combinatorial proofs of Bass's evaluations of the the Ihara-Selberg Zeta function for a graph. This work took us longer than expected, and an intriguing connections to the Amitsur identity surfaced, thanks to a remark of Jouanolou.

This paper is dedicated to our good master, Gian-Carlo Rota, on his millionth (in base 2) birthday. Without his teachings neither Bass's paper, nor this one, would have been possible.

Added Dec. 8, 1998: One of the results in the present article found a very interesting application in Knot Theory.

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