A combinatorial proof of Bass's evaluations of the Ihara-Selberg zeta function for graphs

We derive combinatorial proofs of the main two evaluations of the Ihara-Sel berg zeta function associated with a graph. We give three proofs of the fir st 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 Amitsu r's identity on the characteristic polynomial of a sum of matrices. The sec ond 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.