Abstract
We give two proofs that, for a finite regular graph, the reciprocal of Ihara’s zeta function can be expressed as a simple polynomial times a determinant involving the adjacency matrix of the graph. The first proof is based on representing radial symmetric eigenfunctions on regular trees in terms of certain polynomials. The second proof is a consequence of the fact that the resolvent of the adjacency operator on regular trees is exponential.
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Northshield, S. (1999). Two Proofs of Ihara’s Theorem. In: Hejhal, D.A., Friedman, J., Gutzwiller, M.C., Odlyzko, A.M. (eds) Emerging Applications of Number Theory. The IMA Volumes in Mathematics and its Applications, vol 109. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1544-8_19
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DOI: https://doi.org/10.1007/978-1-4612-1544-8_19
Publisher Name: Springer, New York, NY
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