Combinatorial Computation of Ihara Zeta Functions of Graphs
A simplified combinatorial method for the coefficients of the reciprocal Ihara zeta polynomial of a graph, writing each coefficient as a signed sum over the linear subgraphs (disjoint unions of directed cycles) of the oriented line graph. It yields closed forms for graph families, shows the zeta function is a complete invariant for rank-two graphs, that the polynomial is even exactly for bipartite graphs, and counts spanning trees via its value at u=1.
2501.00639
The reciprocal of the Ihara zeta function of a finite graph is a polynomial graph invariant, computable from a Bass-Hashimoto-type determinant of the graph's adjacency and degree matrices. This paper…