# A combinatorial proof of Bass's determinant formula for the zeta function of regular graphs

## TL;DR

This paper provides a straightforward combinatorial proof of Bass's determinant formula for the zeta function of finite regular graphs, linking cycle counts to eigenvalues via Chebyshev polynomials.

## Contribution

It introduces an elementary combinatorial approach to Bass's formula, connecting non-backtracking cycles with spectral properties of regular graphs.

## Key findings

- Expresses cycle counts using Chebyshev polynomials
- Provides an elementary proof of Bass's determinant formula
- Links spectral eigenvalues to zeta function properties

## Abstract

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 Chebychev polynomials in the eigenvalues of the adjacency operator of the graph.