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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.
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)