Ihara zeta functions and class numbers request pdf. Properties determined by the ihara zeta function of a graph. Such zeta functions have occasionally ap peared in the literature, but mostly through particular cases or under very specific aspects. Random walk on knot diagrams, colored jones polynomial and. This function has proven to be quite fruitful, with applications. Chapter 3 starts with the ruelle zeta function of a. While selbergs zeta function resembles riemanns zeta in some features, there are. The riemann hypothesis is that the nonreal zeros z. On the other hand, their abundance of general explicit properties seems to have been largely ignored, although it can be revealed.
Statistical properties of zeta functions zeros arxiv. We derive combinatorial proofs of the main two evaluations of the iharaselberg zeta function associated with a graph. Chapter 3 starts with the ruelle zeta function of a dynamical system, which. For example, there is a riemann hypothesis which may be false and prime number theorem for graphs. We will be concerned with producing generalizations of many. They are defined by an euler product and have analytic continuation to a meromorphic function with a functional equation. Values of zeta functions and their applications 499 so the same proof gives e cj cfc 3 cfc k 4 even 0 zeta functions the classical zeta functions with arithmetically nice properties arise from algebraic number theory riemann zeta function, dirichlet lseries. Finally there is the ruelle zeta function of a dynamical system. Heat kernels on regular graphs and generalized ihara zeta.
This chapter focuses on selbergiharas zeta function for padic discrete groups. The divisor of selberg s zeta function for kleinian groups patterson, s. That is, we show that this zeta function is the reciprocal of a polynomial which is easily computed if one. In the third proof it is shown that the rst evaluation is an immediate consequence of amitsurs identity on the characteristic polynomial of a sum of matrices. We will consider the riemann hypothesis for the ihara zeta function and its connection with expander graphs. The ihara zeta function may be considered as a modi. The aim of the paper is to formulate a discrete analogue of the claim made by alvarezgaume et al. We give a survey of the results in this area and then give a complete characterization of the coefficients. Soon the connection with graphs was found and many papers appeared. I satisfies the riemann hypothesis precisely when g is ramanujan.
Kiichiro hashimoto and akira hori, selberg ihara s zeta function for padic discrete groups, automorphic forms and geometry of arithmetic varieties, adv. Artinmazur zeta function ruelle zeta function ihara zeta function of a graph airy zeta function, related to the zeros of the airy function. Zeta functions of graphs were studied not only by ihara 34 35, but. Lapidus 1 sep 2008 journal of functional analysis, vol. Of the iharaselberg zeta function for graphs dominique foata and doron zeilberger this paper is dedicated to giancarlo rota, on his millionth2s birthday. Kiichiro hashimoto and akira hori, selbergiharas zeta function for padic discrete groups, automorphic forms and geometry of arithmetic varieties, adv. The ihara zeta function was rstly dened by the aforementioned ihara studying discrete subgroups of the twobytwo special linear groups. Selbergs zeta function of a riemann surface, iharas zeta function of a. Ihara zeta functions for periodic simple graphs springerlink. Theiharaselbergzetafunctionforpgl heckeoperators arxiv. H 2 \displaystyle \gamma \backslash \mathbb h 2, where. In the end, one obtains determinantal formulas for ihara zeta like functions. Introduction the aim of this paper is to give a nontrivial generalization of the ihara selberg zeta function to hypergraphs and show how our generalization can be thought of as a zeta function on a graph.
Keywords and phrases nonbacktracking, ihara zeta, chebyshev polynomial. It closely resembles the so called selberg zeta function, being used to relate closed paths to the spectrum of the adjacency matrix. Ag tdg,wherei is the identity matrix and dg is the diagonal degree matrix of g. Analogies abound with numbertheoretic functions such as riemanndedekind zeta functions. The divisor of selbergs zeta function for kleinian groups patterson, s. As an application, we give a very poor bound on the number of eulerian circuits in a graph. Selbergiharas zeta function for padic discrete groups. We give three proofs of the rst evaluation all based on the algebra of lyndon words. Here we apply the formula to deduce the basic fact about the ihara zeta function of a. A combinatorial proof of basss evaluations of the iharaselberg zeta function for graphs, trans. In this context, the inverse of the alexander polynomial of a knot plays the role of an ihara selberg zeta function of a directed weighted graph, counting with weights cycles of random walk on a 1string link whose closure is the knot in question. The ihara zeta function is considered a padic and a graphtheoretic analogue of the selberg zeta function. Fredholm determinant, ihara zeta function, weighted graphs.
Seiberg duality, quiver gauge theories, and iharas zeta function. Ihara zeta function igor prokhorenkov riemann zeta function recall that the riemann zeta function s satisfies s n 1 n s, re s 1, and there is an analytic continuation. Selberg type zeta functions and arithmetical zeta functions. Pdf poles of the \it ihara zeta function associated with a finite graph are described by. Such function was rst dened in terms of discrete subgroups. In this paper we give an approximation to an ihara formula in the case of the group pgl3. There is no iharaformula for higher rank up to date. Selbergtype zeta functions and arithmetical zeta functions. It closely resembles the selberg zeta function, and is used to relate closed walks to the spectrum of the adjacency matrix.
This formula expresses the zeta function in terms of heckeoperators. On the ihara zeta function of cones over regular graphs. This procedure is motivated by mckean mck72 in his approach to the selberg zeta function and was axiomatized in jl01 to abstract settings. Iis of the form 1 where periodic orbits are replaced by cycles circuits on g without immediate backtracking. In this article, we give a factorization formula for the ihara zeta function of the corona of two graphs, derived from bass determinant formula of the ihara zeta function. In, the ihara zeta function is obtained for several generalizations of the corona construction e.
The zeta functions associated to finite graphs by ihara 7, hashimoto 5, and others combine features of riemanns zeta function, artin lfunctions, and selbergs zeta function. The theory of ihara zeta functions is extended to infinite graphs which are weighted and of finite total. A combinatorial proof of iharabasss formula for the zeta function. Iharaselberg zeta function, lyndon words, amitsur identity. Thesis, czarneski began a preliminary study of the coefficients of the reciprocal of the ihara zeta function of a finite graph. Ihara s zeta function for periodic graphs and its approximation in the amenable case daniele guido, tommaso isola and michel l. Request pdf the iharaselberg zeta function for pgl3 and hecke operators we prove an approximation to an ihara formula for the zeta function of an arithmetic quotient of the bruhattits. A model of random walk on knot diagrams is used to study the alexander polynomial and the colored jones polynomial of knots. Request pdf the iharaselberg zeta function for pgl3 and hecke operators we prove an approximation to an iharaformula for the zeta function of an arithmetic quotient of the bruhattits. Bass, the iharaselberg zeta function of a tree lattice, internat. Oct 17, 2012 in this paper we derive a formula of the ihara zeta function of a cone over a regular graph that involves the spectrum of the adjacency matrix of the cone. We study iharas zeta function for graphs in the context of quivers arising from gauge theories, especially under seiberg duality transformations. There is no ihara formula for higher rank up to date. Ihara selberg zeta function associated with a graph.
The ihara vertex zeta function of x is defined at u. Selberg zetafunction for the modular group edit for the case where the surface is. In this context, the inverse of the alexander polynomial of a knot plays the role of an iharaselberg zeta function of a directed weighted graph, counting with weights cycles of random walk on a 1string link whose closure is the knot in question. The riemann hypothesis is that the nonreal zeros z of s satisfy re z 1 2. Linear algebra and its applications connecting repositories. Values of zeta functions and their applications 499 so the same proof gives e cj cfc 3 cfc k 4 even 0 basssformulafor thezetafunctionofregulargraphs. Riemann, dedekind, selberg, and ihara zetas audrey terras crm montreal 2009 more details can be found in my webpage. In mathematics, the ihara zeta function is a zeta function associated with a finite graph. Iharas zeta function for periodic graphs and its approximation in the amenable case daniele guido, tommaso isola and michel l. Introduction the aim of this paper is to give a nontrivial generalization of the iharaselberg zeta function to hypergraphs and show how our generalization can be thought of as a zeta function on a graph. What are zeta functions of graphs and what are they good for. Theta and selberg zeta function ulrich bunke june 22, 2015 abstract contents 1 the theta function 1 2 hyperbolic space 5 3 the selberg trace formula 8.
Pdf discrete dirac operators, critical embeddings and ihara. Combinatorial analysis by the ihara zeta function of graphs. Selberg defined a zeta function in 1956 see 26 that is associated to a compact. A weak version of the ihara formula is proved for zeta functions attached to quotients of the bruhattits building of pgl3. Ihara zeta functions of finite graphs are reciprocals of polynomials, sometimes in several variables. Indeed, it is a zeta function associated with a nite graph.
978 1182 439 1352 1391 1314 667 972 319 1135 69 757 300 990 1249 1304 832 1386 1059 882 406 1366 1435 945 1461 525 457 12 576 130 1162 173 1271 549 939 297 51 550 700 666 367