Zeta functions of heisenberg graphs over finite rings. In the case of circular graphs, the laplace zeta function is n2s. Moreover,we constructa new unboundedfamily of ramanujan. One example of this, for m 1, is given by the birchswinnertondyer formula 5, since the integers s and. Relating zeta functions of discrete and quantum graphs. A survey on inclusion properties among relations for multiple zeta values kazuhiro yoshikawa ritsumeikan univ. These zeta functions are geometrically motivated, but can be viewed as certain generalizations of characteristic polynomials. Quantum graphs graph zeta functions discrete verses quantum zeta functions zeta functions of quantum graphs jon harrison1 gregory berkolaiko2, klaus kirsten1, uzy smilansky3. In mathematics, the ihara zeta function is a zeta function associated with a finite graph. If you want to merge data from two graphs, rather than create a new graph from scratch, you can superimpose the two using a simple copy and paste operation. We will discuss several properties of finite graphs that are determined by the zeta function and show by counterexample several.
The aim was to give a short introduction on zeta functions over. Recall that a cycle in a graph is oriented and has a starting point. This allows us to give explicit formulae for the zeta function of xn for small n. The ihara zeta function of a finite graph reflects combinatorial and spectral properties of that graph 2,12,16. The riemann zeta function david jekel june 6, 20 in 1859, bernhard riemann published an eightpage paper, in which he estimated \the number of prime numbers less than a given magnitude using a certain meromorphic function. The riemann zeta function riemanns zeta function for s 2c with re.
Note that the restriction on the convergence of the function does not stop us from relating any two values of the zeta function however, the function also admits an analytic continuation. To prove the result we actually look at the dirac zeta function ns p 0 swhere are the positive eigenvalues of the dirac operator of the circular graph. The idea was generalized to all finite graphs in 1989 by hashimoto. We explore three seemingly disparate but related avenues of inquiry. Along the way, both concrete examples and open problems are presented to illustrate the general theory. Elementary proofs based on the notions of \it oriented line graphs, \it perron. Pdf weighted zeta functions for quotients of regular. The ihara zeta function for graphs and 3adic convergence. After discussing some graph properties determined by zeta functions, we show that large classes of cospectral graphs can be distinguished with zeta functions and enumerate graphs distinguished by zeta functions on 11. The zeta functions associated to finite graphs by ihara 7, hashimoto 5, and others combine features of riemanns zeta function, artin l functions, and selbergs zeta function. We will introduce the zeta function from both perspectives and show the equivalence of both forms. If the graphs in the set have no blank nodes in common, then the union of the graphs is a merge. It closely resembles the selberg zeta function, and is used to relate closed walks to the spectrum of the adjacency matrix.
Extended summary the zeta function gs for a nite simple graph. Zeta graphs are related to multiplevalued zeta functions, and we show that the expectation of a conjunctive query can be written as a linear combination of multiplevalued zeta functions. Multiple zeta functions, multiple polylogarithms and their special values. What they all have in common is that they are complex functions. Dec 16, 20 as the zeta functions zeta n,s of the circular graphs are entire functions, the result does not say anything about the roots of the classical riemann zeta function zeta s, which is also the dirac zeta function for the circle. On analytic continuation of various multiple zeta functions kohji matsumoto abstract in this article we describe the development of the problem of analytic continuation of multiple zeta functions. Zeta functions of graphs 3 we do not consider zeta functions of in. To do this, we apply a relation between the spectrum of the laplacian on a discrete graph and that of the laplacian on an equilateral metric graph. Zeta functions of finite graphs and converings, part ii. The aim of these lectures is to provide an intorduc tion to the theory of the riemann zetafunction for stu dents who might later want to do research on the subject.
Conjecturally, almost all graphs are determined by. They are defined by an euler product and have analytic continuation to a meromorphic function with a functional equation. September 30, 2015 abstract we initiate the study of spectral zeta functions. As an application, we show that these graphs are very good concentrators. The zeta function of a nite graph x counts the number of geodesic tailless cycles in x. Equally good would be an algorithm to traverse all graphs at once. The three zeta functions are associated to one vertex variable, two variables for each edge, and 2r2r. Basic zeta functions and some applications in physics klaus kirsten 1. In order to do this, we must compute the galois group of the covering obtained by reducing coordinates in. The ihara zeta function plays an important role in the study of free groups, spectral graph theory, and dynamical systems, especially symbolic dynamics, where the ihara zeta function is an example of a ruelle zeta function. Notes on the riemann zeta function january 25, 2007 1 the zeta function 1. Zeta functions of graphs were studied not only by ihara 34 35, but. I have tried to put the formula for the riemann zeta function into excel using the figure for the second zero, 0.
The zeta function of a graph may play a role for such a task. Deitmar schemes, graphs and zeta functions sciencedirect. The fact that this function is analytic in this region of the complex plane is a consequence of the following basic fact. Poles of the \it ihara zeta function associated with a finite graph are described by graphtheoretic quantities. Poles of the \it ihara zeta function associated with a finite graph are described by graph theoretic quantities.
Audrey terras describes in zeta functions and chaos three zeta functions the zeta fct. Zeta functions of quantum graphs university of oxford. Such zeta functions are considered, for example, by clair and mokhtarisharghi 5aswellas grigorchuk and zuk 9. Note that two graphs have the same zeta function if and only if they have the same number of divisor classes of each degree and rank. The theory of zeta functions has been extended from the complex plane to number fields, algebraic curves and graphs. Sections 3 and 4 we will consider the multivariable zeta functions known as edge and path zeta functions of graphs. Twovariable zetafunctions on graphs and riemannroch. Note that a regular graph is a ramanujan graph if and only if the ihara zeta function of such graph satises an analogue of riemann hypothesis, translated to graph theory.
The theory of zeta graphs with an application to random. Zeta functions of the dirac operator on quantum graphs. The ihara zeta function of a graph satisfies a generalized form of the riemann hypothesis if it is a ramanujan graph. Zeta functions of finite graphs 9 algebraonxwhichwecannotapplytothegeneralcaseseealso20,21 andj. As a consequence it extends to a meromorphic function on a riemann surface. In the dialog, select specified with the merge dropdown list. Kurokawa defines the f 1 zeta function of a zscheme x of f 1type as 17. Analogies abound with numbertheoretic functions such as riemann or dedekind zeta functions. Though terras already proved this in her paper 5, we will expand the proof to. Here zeta and l functions are reciprocals of polynomials which means these functions. The meromorphic extension provides a setting to generalize known properties of zeta functions of regular graphs. Z c z s d dz log fzdz where c is any contour that encloses the zeros of f while avoiding its poles.
Finally, in section 9, we prove that the zeta function. A merge of a set of rdf graphs is defined as follows. The zeros of this function will be symmetrically distributed around the critical line. Zeta functions of graphs, a stroll through the garden. 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. Combinatorial analysis by the ihara zeta function of graphs. Spectral zeta function fz z xb b1 cos b coszl b sinzl b s 2 x1 j1 0k s j 1 i. Ihara zeta functions of finite graphs are reciprocals of polynomials, sometimes in several variables. Spectral zeta functions of graphs and the riemann zeta function in the critical strip. We investigate iharaselberg zeta functions of cayley graphs for the heisenberg group over finite rings. Mellin, and then discuss the euler sum and its multivariable generalization. Three different zeta functions are attached to a finite connected, possibly irregular graphx.
It is called tailless if it remains a geodesic regardless the initial point. On analytic continuation of various multiple zetafunctions. Infinite divisibility induced by identities of multiple zeta values. Weighted zeta functions for quotients of regular coverings of graphs. The ihara zeta function is defined by the determinant of the adjacency matrix of the graph. In the following example, we merge two graphs graph1 and graph2 into a new graph page. We will show how to specialize the path zeta to the edge zeta and the edge zeta to the original ihara vertex zeta. For simplicity, we have kept the original lecture style of the notes. Zeta functions for graphs in this section we begin by recalling the properties of the ihara zeta function for.
Mathematical and theoretical hwdixqfwlrqvritxdqwxpjudskv to cite this article. In a second way, the spectral zeta function of the graph z enjoys properties analogous to the riemann zeta function, notably the relation. Elementary proofs based on the notions of \it oriented line graphs, \it. What are zeta functions of graphs and what are they good for. Maybe can the symmetric zeta functions give new insights about the riemann hypothesis. Only for res1, the values of zeta n,s converge suitably scaled to zeta s. The connection between f 1 zeta functions and arithmetic zeta functions. The forced size options fysize and fxsize are allowed with any graph. Here we attempt to survey some of the basic theory as presented in stark and terras 28, 29, 30. It is primitive if it is not obtained by tracing a shorter cycle more than once. In particular, two graphs with the same zeta function. In section 8, we establish several functional equations, for di. In section 7, we prove a corresponding determinant formula. A note on jacobians, tutte polynomials, and twovariable.
The ihara zeta function was rstly dened by the aforementioned ihara studying discrete subgroups of the twobytwo special linear groups. Introduction it is the aim of these lectures to introduce some basic zeta functions and their uses in the areas of the. Multiple zeta functions i shinya koyama keio university nobushige kurokawa tokyo institute of technology running title. By merge i mean combining all edges and vertices of both graphs together preserving weights of course, if they do not create cycles. Spectral zeta functions of graphs and the riemann zeta function in the critical strip fabien friedli and anders karlsson. Zeta functions on gr aphs and riemannroch theorems 5 the order of the group pic 0. A graph theoretical analogue of brauersiegel theory for zeta functions of number elds is developed using the theory of artin lfunctions for galois coverings of graphs from parts i and ii. Ihara zeta functions for periodic simple graphs daniele guido, tommaso isola and michel l. The spectra of matrices such as the adjacency and edge adjacency matrices of a graph are essential to the plot of this book, which makes connections with quantum. We gave in 14 a simple proof of the fact that for an unramified graph covering y x the ihara zeta function of x divides the ihara zeta function. Ramanujan graphs are characterized by upper bounds on the eigenvalues of the adjacency matrix. Oct 27, 2017 we write the spectral zeta function of the laplace operator on an equilateral metric graph in terms of the spectral zeta function of the normalized laplace operator on the corresponding discrete graph.
Originally, ihara defined the zeta function on finite graphs imitating the classical definition of the zeta function. Basic zeta functions and some applications in physics. Zeta functions of finite graphs and coverings sciencedirect. In this paper we explore an interesting relationship between discretetime quantum walks and the ihara zeta function of a graph. Zeta mahler measures, multiple zeta values and lvalues tatsushi tanaka kyoto sangyo univ. Twovariable zetafunctions on graphs and riemannroch theorems. The prime number theorem, hardys theorem on the zeros of. In the new graph, the source graphs are arranged in a row by col grid. Arithmetical graphs, with g 0g, m, r1,2, and 3, have been completely enumerated 25, 31. It closely resembles the so called selberg zeta function, being used to relate closed paths to the spectrum of the adjacency matrix. Values of zeta functions and their applications 505 essentially a square when m h m, i. This functional equation relates the zeta functions. Graph theory meets number theory in this stimulating book.
We consider the ihara zeta functions of line graphs, middle graphs and total graphs of a regular graph and their regular or irregular covering graphs. They are defined by an euler product and have analytic continuation to a meromorphic function. Zeta functions of graphs graph theory meets number theory in this stimulating book. The second is a draft of a book on zeta functions of graphs.
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