Zeta Functions Of Graphs Research Articles

Graph Zeta function and gauge theories

Along the recently trodden path of studying certain number theoretic properties of gauge theories, especially supersymmetric theories whose vacuum manifolds are non-trivial, we investigate Ihara's Graph Zeta Function for large classes of quiver theories and periodic tilings by bi-partite graphs. In particular, we examine issues such as the spectra of the adjacency and whether the gauge theory satisfies the strong and weak versions of the graph theoretical analogue of the Riemann Hypothesis.

Zeta functions of graphs with [formula omitted] actions

Suppose Y is a regular covering of a finite graph X with covering transformation group π = Z . This paper gives an explicit formula for the L 2 zeta function of Y and computes examples. When π = Z , the L 2 zeta function is an algebraic function. As a consequence it extends to a meromorphic function on a Riemann surface. The meromorphic extension provides a setting to generalize known properties of zeta functions of regular graphs, such as the location of singularities and the functional equation.

A generalized characteristic polynomial of a graph having a semifree action

For an abelian group Γ , a formula to compute the characteristic polynomial of a Γ -graph has been obtained by Lee and Kim [Characteristic polynomials of graphs having a semi-free action, Linear algebra Appl. 307 (2005) 35–46]. As a continuation of this work, we give a computational formula for generalized characteristic polynomial of a Γ -graph when Γ is a finite group. Moreover, after showing that the reciprocal of the Bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph, we compute the reciprocals of the Bartholdi zeta functions of wheels and complete bipartite graphs as an application of our formula.

Weighted Bartholdi zeta functions of graphs

We define the weighted Bartholdi zeta function of a graph G , and give a determinant expression of it. Furthermore, we define a weighted L -function of G , and present a determinant expression for the weighted L -function of G . As a corollary, we show that the weighted Bartholdi zeta function of a regular covering of G is a product of weighted L -functions of G .

Weighted zeta functions of graphs

We give a decomposition formula for the weighted zeta function of a regular covering of a graph.

Decomposition formulas of zeta functions of graphs and digraphs

We give a decomposition formula of the zeta function of a regular covering of a graph G with respect to equivalence classes of prime, reduced cycles of G. Furthermore, we give a decomposition formula of the zeta function of a g-cyclic Γ-cover of a symmetric digraph D with respect to equivalence classes of prime cycles of D, for any finite group Γ and g∈ Γ.

Convergence of zeta functions of graphs

The $L^{2}$-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 $L^{2}$-zeta function of $Y$.

Finite Quantum Chaos

Quantum chaos is in part the study of the statistics of energy levels of a quantum system; i.e., the eigenvalues of the Schrodinger operator. Thus physicists began to investigate the spectra of random large matrices. Similarly number theorists and geometers have investigated the statistics of spectra of Laplacians on Riemannian manifolds which is equivalent to studying the statistics of the zeros of the Selberg zeta function. Parallels with the statistics of the zeros of the Riemann zeta function quickly became evident. Here we consider nite analogs in which the zeta function is the Ihara zeta function of a graph (and zeros are replaced by poles). The picture in the PDF abstract le shows all poles but -1 of the Ihara zeta function for a random irregular graph with 80 vertices. In this context, the Riemann hypothesis says the poles lie outside the green circle. A reference for this talk is my book Zeta Functions of Graphs.

Finite Quantum Chaos

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsAudrey TerrasAUDREY TERRAS received her B.S. degree in Mathematics from the University of Maryland, College Park in 1964, where she was inspired by the lectures of Sigekatu Kuroda to become a number theorist. She was particularly impressed by the use of analysis (in particular using zeta functions) to derive algebraic results. She received her M.A. (1966) and Ph.D. (1970) from Yale University.In 1972 she became an assistant professor of mathematics at the University of California, San Diego. Now she is a full professor at U.C.S.D. where she has had 21 Ph.D. students. She is a fellow of the American Association for the Advancement of Science, and has served on the Council of the American Mathematical Society.Her research interests include number theory, harmonic analysis on symmetric spaces and finite groups along with its applications, special functions, algebraic graph theory, especially zeta functions of graphs, and Selberg's trace formula. When lecturing on mathematics she believes it is important to give examples, applications, and colorful pictures.