A graph theoretical analogue of Brauer-Siegel theory for zeta functions of number …elds is developed using the theory of Artin L-functions for Galois coverings of graphs from parts I and II. In the process, we discuss possible versions of the Riemann hypothesis for the Ihara zeta function of an irregular graph.
The goal of this book is to guide the reader in a stroll through the garden of zeta functions of graphs. The subject arose in the late part of the last century modelled after zetas found in the other gardens. Number theory involves many zetas starting with Riemann's-a necessary ingredient in the study of the distribution of prime numbers. Other zetas of… (More)
which satisfy suitable initial or boundary conditions are known to exhibit a maximum principle property. A detailed study of this phenomenon (mostly in the two variables case) is the subject matter of the last chapter. In conclusion, the book gives a very readable account of the role of maximum principles in differential equations. It should be read by… (More)
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.
We present examples of hypergraphs constructed from homogeneous spaces of finite general linear groups. These hypergraphs are constructed using an invariant analogue of a hypervolume and their spectra are analyzed to see if they are Ramanujan in the sense of W.-C.
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)
2000 iii To m y Mother and Father We could use up two Eternities in learning all that is to be l e arned a b out our own world and the thousands of nations that have arisen and ourished and vanished f r om it. Mathematics alone would occupy me eight million years. |Mark Twain.