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We derive combinatorial proofs of the main two evaluations of theIhara-Selberg Zeta function associated with a graph. We give three proofs ofthe first evaluation all based on the algebra of Lyndon words. In the thirdproof it is shown that the first evaluation is an immediate consequence ofAmitsur's identity on the characteristic polynomial of a sum of matrices. Thesecond evaluation of the Ihara-Selberg Zeta function is first derived by meansof a sign-changing involution technique. Our second approach makes use of ashort matrix-algebra argument.

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