It closely resembles the so called selberg zeta function, being used to relate closed paths to the spectrum of the adjacency matrix. Finally there is the ruelle zeta function of a dynamical system. Pdf poles of the \it ihara zeta function associated with a finite graph are described by. In, the ihara zeta function is obtained for several generalizations of the corona construction e.
Such function was rst dened in terms of discrete subgroups. The ihara vertex zeta function of x is defined at u. Iis of the form 1 where periodic orbits are replaced by cycles circuits on g without immediate backtracking. Keywords and phrases nonbacktracking, ihara zeta, chebyshev polynomial. Ag tdg,wherei is the identity matrix and dg is the diagonal degree matrix of g. Indeed, it is a zeta function associated with a nite graph. We also have s p prime 1 p s 1 exercise the functional equation is s s2 s 2 s 1 s. In this paper we give an approximation to an ihara formula in the case of the group pgl3. 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. Ihara zeta functions of finite graphs are reciprocals of polynomials, sometimes in several variables.
Artinmazur zeta function ruelle zeta function ihara zeta function of a graph airy zeta function, related to the zeros of the airy function. The riemann hypothesis is that the nonreal zeros z. Analogies abound with numbertheoretic functions such as riemanndedekind zeta functions. Kiichiro hashimoto and akira hori, selberg ihara s zeta function for padic discrete groups, automorphic forms and geometry of arithmetic varieties, adv. Ihara selberg zeta function associated with a graph. We give three proofs of the rst evaluation all based on the algebra of lyndon words. Selberg defined a zeta function in 1956 see 26 that is associated to a compact. Values of zeta functions and their applications 499 so the same proof gives e cj cfc 3 cfc k 4 even 0 basssformulafor thezetafunctionofregulargraphs. That is, we show that this zeta function is the reciprocal of a polynomial which is easily computed if one. 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. Fredholm determinant, ihara zeta function, weighted graphs. 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. In the end, one obtains determinantal formulas for ihara zeta like functions.
The divisor of selbergs zeta function for kleinian groups patterson, s. 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. Lapidus 1 sep 2008 journal of functional analysis, vol. Here we apply the formula to deduce the basic fact about the ihara zeta function of a.
The divisor of selberg s zeta function for kleinian groups patterson, s. We give a survey of the results in this area and then give a complete characterization of the coefficients. Iharaselberg zeta function, lyndon words, amitsur identity. This chapter focuses on selbergiharas zeta function for padic discrete groups. 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. Selbergtype zeta functions and arithmetical zeta functions. Bass, the iharaselberg zeta function of a tree lattice, internat. Kiichiro hashimoto and akira hori, selbergiharas zeta function for padic discrete groups, automorphic forms and geometry of arithmetic varieties, adv. Thesis, czarneski began a preliminary study of the coefficients of the reciprocal of the ihara zeta function of a finite graph. Selberg zetafunction for the modular group edit for the case where the surface is.
Of the iharaselberg zeta function for graphs dominique foata and doron zeilberger this paper is dedicated to giancarlo rota, on his millionth2s birthday. Iharas zeta function for periodic graphs and its approximation in the amenable case daniele guido, tommaso isola and michel l. The riemann hypothesis is that the nonreal zeros z of s satisfy re z 1 2. H 2 \displaystyle \gamma \backslash \mathbb h 2, where. 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. What are zeta functions of graphs and what are they good for. A combinatorial proof of basss evaluations of the iharaselberg zeta function for graphs, trans. They are defined by an euler product and have analytic continuation to a meromorphic function with a functional equation. Chapter 3 starts with the ruelle zeta function of a dynamical system, which. A combinatorial proof of iharabasss formula for the zeta function. Linear algebra and its applications connecting repositories.
We will consider the riemann hypothesis for the ihara zeta function and its connection with expander graphs. 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. 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. Ihara zeta functions for periodic simple graphs springerlink. The ihara zeta function is considered a padic and a graphtheoretic analogue of the selberg zeta function. Pdf discrete dirac operators, critical embeddings and ihara. Combinatorial analysis by the ihara zeta function of graphs. Chapter 3 starts with the ruelle zeta function of a.
There is no iharaformula for higher rank up to date. 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. A weak version of the ihara formula is proved for zeta functions attached to quotients of the bruhattits building of pgl3. It closely resembles the selberg zeta function, and is used to relate closed walks to the spectrum of the adjacency matrix. On the ihara zeta function of cones over regular graphs. A model of random walk on knot diagrams is used to study the alexander polynomial and the colored jones polynomial of knots. Theiharaselbergzetafunctionforpgl heckeoperators arxiv. On the other hand, their abundance of general explicit properties seems to have been largely ignored, although it can be revealed. 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. We study iharas zeta function for graphs in the context of quivers arising from gauge theories, especially under seiberg duality transformations. Selberg type zeta functions and arithmetical zeta functions.
Selbergiharas zeta function for padic discrete groups. This formula expresses the zeta function in terms of heckeoperators. For example, there is a riemann hypothesis which may be false and prime number theorem for graphs. Statistical properties of zeta functions zeros arxiv.
We will be concerned with producing generalizations of many. Selbergs zeta function of a riemann surface, iharas zeta function of a. Dynamical zeta functions and the distribution of orbits. As an application, we give a very poor bound on the number of eulerian circuits in a graph. Zeta functions of graphs were studied not only by ihara 34 35, but. Selberg zeta function of a riemann surface witten zeta function of a lie group spectral zeta function of an operator dynamical systems. I satisfies the riemann hypothesis precisely when g is ramanujan. The ihara zeta function may be considered as a modi. We derive combinatorial proofs of the main two evaluations of the iharaselberg zeta function associated with a graph. There is no ihara formula for higher rank up to date. Such zeta functions have occasionally ap peared in the literature, but mostly through particular cases or under very specific aspects. This procedure is motivated by mckean mck72 in his approach to the selberg zeta function and was axiomatized in jl01 to abstract settings. The ihara zeta function was rstly dened by the aforementioned ihara studying discrete subgroups of the twobytwo special linear groups.
The aim of the paper is to formulate a discrete analogue of the claim made by alvarezgaume et al. Ihara s zeta function for periodic graphs and its approximation in the amenable case daniele guido, tommaso isola and michel l. Random walk on knot diagrams, colored jones polynomial and. Ihara zeta functions and class numbers request pdf. Properties determined by the ihara zeta function of a graph. In mathematics, the ihara zeta function is a zeta function associated with a finite graph. 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. Heat kernels on regular graphs and generalized ihara zeta.
This function has proven to be quite fruitful, with applications. 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. We show that the ihara zeta function and the spectrum of the adjacency matrix of the cone determine each other and we characterize those cones that satisfy the graph theory riemann hypothesis. Seiberg duality, quiver gauge theories, and iharas zeta function. The theory of ihara zeta functions is extended to infinite graphs which are weighted and of finite total. We derive combinatorial proofs of the main two evaluations of the iharaselberg zeta function associated with. Selbergs zeta function of a riemann surface, iharas zeta function of a finite connected graph. Soon the connection with graphs was found and many papers appeared. 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.
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