Alladi Ramakrishnan Hall

A Combinatorial Proof of Ihara-Bass's Formula for the Zeta Function

Bharatram Rangarajan

Tel Aviv University

We give an elementary combinatorial proof of Bass's determinant
formula for the zeta function of a finite regular graph. This is done by
expressing the number of non-backtracking cycles of a given length in terms
of Chebyshev polynomials in the eigenvalues of the adjacency operator of the
graph. A related observation of independent interest is that the Ramanujan
property of a regular graph is equivalent to tight bounds on the number of
non-backtracking cycles of every length.