Zeta Functions of Finite Graphs
نویسندگان
چکیده
Poles of the Ihara zeta function associated with a finite graph are described by graph-theoretic quantities. Elementary proofs based on the notions of oriented line graphs, Perron-Frobenius operators, and discrete Laplacians are provided for Bass’s theorem on the determinant expression of the zeta function and Hashimoto’s theorems on the pole at u = 1.
منابع مشابه
Convergence of Zeta Functions of Graphs
The L-zeta function of an infinite graph Y (defined previously in a ball around zero) has an analytic extension. For a tower of finite graphs covered by Y , the normalized zeta functions of the finite graphs converge to the L-zeta function of Y . Introduction Associated to any finite graph X there is a zeta function Z(X,u), u ∈ C. It is defined as an infinite product but shown (in various diffe...
متن کاملWhat are zeta functions of graphs and what are they good for ? Matthew
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.
متن کاملZeta Functions of Finite Graphs and Coverings, Iii
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.
متن کاملSpectral zeta functions of graphs and the Riemann zeta function in the critical strip
We initiate the study of spectral zeta functions ⇣X for finite and infinite graphs X, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann’s zeta funct...
متن کاملZeta Functions of Infinite Graph Bundles
We compute the equivariant zeta function for bundles over infinite graphs and for infinite covers. In particular, we give a “transfer formula” for the zeta function of infinite graph covers. Also, when the infinite cover is given as a limit of finite covers, we give a formula for the limit of the zeta functions.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2004