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 walks 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.
Definition.
The Ihara zeta function is defined as the analytic continuation of the infinite product
formula_1
The product in the definition is taken over all prime closed geodesics formula_2 of the graph formula_3, where geodesics which differ by a cyclic rotation are considered equal. A "closed geodesic" formula_2 on formula_5 (known in graph theory as a "closed walk") is a finite sequence of vertices formula_6 such that
The integer formula_9 is the "length" formula_10 of formula_2. The closed geodesic formula_2 is "prime" if it cannot be obtained by repeating a closed geodesic formula_13 times, for an integer formula_14.
This graph-theoretic formulation is due to Sunada.
Ihara's formula.
Ihara (and Sunada in the graph-theoretic setting) showed that for regular graphs the zeta function is a rational function.
If formula_5 is a formula_16-regular graph with adjacency matrix formula_17 then
where formula_19 is the circuit rank of formula_5. If formula_5 is connected and has formula_22 vertices, formula_23.
The Ihara zeta-function is in fact always the reciprocal of a graph polynomial:
where formula_25 is Ki-ichiro Hashimoto's edge adjacency operator. Hyman Bass gave a determinant formula involving the adjacency operator.
Applications.
The Ihara zeta function plays an important role in the study of free groups, spectral graph theory, and dynamical systems, especially symbolic dynamics, where the Ihara zeta function is an example of a Ruelle zeta function.