Efficient computation of Ihara coefficients using the Bell polynomial recursion

The Ihara zeta function has proved to be a powerful tool in the analysis of graph structures. It is determined by the prime cycles of a finite graph G=(V,E) and can be characterized in terms of a quasi characteristic polynomial of the adjacency matrix T of the oriented line graph associated to G. The coefficients of this polynomial, referred to as Ihara coefficients, have been used to characterize graphs in a permutation-invariant manner, and allow for an efficient evaluation of the Ihara zeta function. In this paper we present a novel method for computing the Ihara coefficients. We first establish a characterization of the Ihara coefficients in terms of complete Bell polynomials and, by exploiting a recursive relation for the latter, we show how the Ihara coefficients can be efficiently computed in O(|E|2), provided that the eigenvalues of T are known.