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. 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-functions product is taken over all prime closed geodesics p {displaystyle p} of the graph G = ( V , E ) {displaystyle G=(V,E)} , where a closed geodesic p {displaystyle p} on G {displaystyle G} is a finite sequence of vertices p = ( e 0 , … , e k − 1 ) {displaystyle p=(e_{0},ldots ,e_{k-1})} such that: The integer k {displaystyle k} is the length L ( p ) {displaystyle L(p)} of p {displaystyle p} . The closed geodesic p {displaystyle p} is prime if it cannot be obtained by repeating a closed geodesic m {displaystyle m} times, for an integer m > 1 {displaystyle m>1} .