Abstract
The infinite grid is the Cayley graph of $\mathbb{Z} \times \mathbb{Z}$ with the usual generators. In this paper, the Ihara zeta function for the infinite grid is computed using elliptic integrals and theta functions. The zeta function of the grid extends to an analytic, multivalued function which satisfies a functional equation. The set of singularities in its domain is finite.The grid zeta function is the first computed example which is non-elementary, and which takes infinitely many values at each point of its domain. It is also the limiting value of the normalized sequence of Ihara zeta functions for square grid graphs and torus graphs.
Highlights
Graph theory zeta functions first appeared in the 1960’s, introduced by Ihara and later refined by Serre, Hashimoto, and Bass
Like the Riemann zeta function, the zeta function of a graph is an Euler product taken over primes
The Euler product only converges on a small open set, and analytic continuation to the plane is performed by a functional equation
Summary
Graph theory zeta functions first appeared in the 1960’s, introduced by Ihara and later refined by Serre, Hashimoto, and Bass. In the case of graphs, the primes are certain closed paths, analogous to the geodesics used as primes in Selberg’s zeta function for a hyperbolic surface For these zeta functions, the Euler product only converges on a small open set, and analytic continuation to the plane is performed by a functional equation. The Euler product definition of the zeta function for infinite graphs mirrors closely the definition for finite graphs, and there is an analog of the Ihara formula giving the zeta function in terms of the determinant of a Laplacian operator. With finite graphs, this operator acts on the finite dimensional space of functions on vertices, and so the determinant is a polynomial.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
