#### 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.

Done