Several Proofs of Ihara's Theorem
نویسنده
چکیده
We give three proofs that 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 rst proof, for regular graphs, is based on representing radial symmetric eigenfunctions on regular trees in terms of certain polynomials. The second proof, also for regular graphs, is a consequence of the fact that the resolvent of the adjacency operator on regular trees is exponential. A third proof,in many ways simpler than the rest, works for irregular graphs as well.
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Two Proofs of Ihara's Theorem
We give two proofs that, for a nite 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 rst 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 adjacenc...
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