Ihara Zeta Function

Ihara Zeta Function

In mathematics, the Ihara zeta-function is a zeta function associated with a finite graph. It closely resembles the Selberg zeta-function, and is used to relate closed paths to the spectrum of the adjacency matrix. The Ihara zeta-function 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 suggested in his book Trees that Ihara's original definition can be reinterpreted graph-theoretically. It was Toshikazu Sunada who put this suggestion into practice (1985). A regular graph is a Ramanujan graph if and only if its Ihara zeta function satisfies an analogue of the Riemann hypothesis.