Ihara Zeta Function

In mathematics, the Ihara zeta function is a zeta function associated with a finite graph, closely resembling the Selberg zeta function and used to relate closed walks in the graph to the spectrum of its adjacency matrix. It was first defined by Yasutaka Ihara in the 1960s in the context of discrete subgroups of the two-by-two p-adic special linear group; Jean-Pierre Serre later suggested Ihara's definition could be reinterpreted graph-theoretically, and Toshikazu Sunada carried that out in 1985. As Sunada observed, a regular graph is a Ramanujan graph if and only if its Ihara zeta function satisfies an analogue of the Riemann hypothesis.