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 in 1985. As observed by Sunada, a regular graph is a Ramanujan graph if and only if its Ihara zeta function satisfies an analogue of the Riemann hypothesis.

The Ihara zeta-function can be defined by a formula analogous to the Euler product for the Riemann zeta function:

This product is taken over all prime walks p of the graph ${\displaystyle G=(V,E)}$ – that is, closed cycles ${\displaystyle p=(u_{0},\ldots ,u_{L(p)-1},u_{0})}$ such that

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