منابع مشابه
Several Proofs of Ihara's Theorem
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 res...
متن کاملTwo 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...
متن کاملEuclidean Proofs of Dirichlet’s Theorem
Euclid’s proof of the infinitude of the primes is a paragon of simplicity: given a finite list of primes, multiply them together and add one. The resulting number, say N , is not divisible by any prime on the list, so any prime factor of N is a new prime. Some special cases of Dirichlet’s theorem admit a simple proof following Euclid’s model, such as the case of 1 mod 4 or 5 mod 6. (We mean by ...
متن کاملTwo Proofs of Cayley’s Theorem
We present two proofs of the celebrated Cayley theorem that the number of spanning trees of a complete graph on n vertices is nn−2. In this expository note we present two proofs of Cayley’s theorem that are not as popular as they deserve to be. To set up the story we revisit first some terminology. By a graph G we mean a pair (V (G), E(G)), where V (G) is a set of points (or vertices) and E(G) ...
متن کاملSimple Proof of the Prime Number Theorem , etc
The point here is the relatively simple argument that non-vanishing of an L-function on the line Re (s) = 1 implies an asymptotic result parallel to the application of ζ(s) to the Prime Number Theorem. This is based upon [Newman 1980]. In particular, this argument avoids estimates on the zeta function at infinity and also avoids Tauberian arguments. For completeness, we recall the standard clev...
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ژورنال
عنوان ژورنال: The Mathematical Gazette
سال: 1905
ISSN: 0025-5572
DOI: 10.2307/3603880