Geodesic
The Ihara zeta function of the infinite grid
Bryan Clair1
1Saint Louis University
The electronic journal of combinatorics, Tome 21 (2014) no. 2
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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.
DOI : 10.37236/3561
Classification : 11M41, 05C25
Mots-clés : graph zeta functions, square grid graph, periodic graphs

Bryan Clair  1

1 Saint Louis University 
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Bryan Clair. The Ihara zeta function of the infinite grid. The electronic journal of combinatorics, Tome 21 (2014) no. 2. doi: 10.37236/3561

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