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We discuss zeta functions of finite irregular undirected connected graphs (which may be weighted) and apply them to obtain, for example an analog of the prime number theorem for cycles in graphs. We consider 3 types of zeta functions of graphs: vertex, edge, and path. Analogs of the Riemann hypothesis are also introduced.
put forward several conjectures that continue to occupy us to this day. Gauss stated his conjectures in the language of binary quadratic forms (of even discriminant only, a complication that was later dispensed with). Since Dedekind's time, these conjectures have been phrased in the language of quadratic fields. This is how we will state the conjectures(More)
It is well known that every Euclidean ring is a principal ideal ring. It is also known for a very long time that the converse is not valid. Counterexamples exist under the rings R of integral algebraic numbers in quadratic complex fields Q √ −D , for D = 19, 43, 67, and 163. In conection with these counterexamples several results were published in an effort(More)
We investigate Ihara-Selberg zeta functions of Cayley graphs for the Heisenberg group over finite rings Z/p n Z, where p is a prime. In order to do this, we must compute the Galois group of the covering obtained by reducing coordinates in Z/p n+1 Z modulo p n. The Ihara-Selberg zeta functions of the Heisenberg graph mod p n+1 factor as a product of Artin(More)