Artin Type L-Functions and the Density Theorem for Prime Cycles on Finite Graphs

Hashimoto, Ki-ichiro

doi:10.1142/s0129167x92000370

Cited by 51 publications

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“…Let us mention that other number-theoretic properties, like a version of the Riemann hypothesis, have been studied for the Ihara zeta function, the graphs satisfying it being completely characterized; see [39] for a simple proof. More applications of the Ihara zeta function are contained in [4,21,22,38,34,40,41,23]. Different generalizations of the Ihara zeta function are considered in the literature; see [3,31] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

A trace on fractal graphs and the Ihara zeta function

Guido

Isola

Lapidus

2008

Trans. Amer. Math. Soc.

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Abstract. Starting with Ihara's work in 1968, there has been a growing interest in the study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and Sato, to name just a few authors. Then, Clair and Mokhtari-Sharghi studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms. The main formula in all these treatments establishes a connection between the zeta function, originally defined as an infinite product, and the Laplacian of the graph. In this article, we consider a different class of infinite graphs. They are fractal graphs, i.e. they enjoy a selfsimilarity property. We define a zeta function for these graphs and, using the machinery of operator algebras, we prove a determinant formula, which relates the zeta function with the Laplacian of the graph. We also prove functional equations, and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs. IntroductionThe Ihara zeta function, originally associated to certain groups and then combinatorially reinterpreted as associated with finite graphs or with their infinite coverings, is defined here for a new class of infinite graphs, called self-similar graphs. The corresponding determinant formula and functional equations are established.The combinatorial nature of the Ihara zeta function was first observed by Serre (see [37], Introduction), but it was only through the works of Sunada [42], Hashimoto [19,20] and Bass [4] that it became a graph-theoretical object, at the same time keeping some number-theoretically flavoured properties, such as the Euler product formula or the functional equation.The Ihara zeta function [24] was written as an infinite product (Euler product) over G-conjugacy classes of primitive elements in a group G, namely elements whose centralizer (in G) is generated by the element itself. As explained in detail in the Key words and phrases. Self-similar fractal graphs, Ihara zeta function, geometric operators, C*-algebra, analytic determinant, determinant formula, primitive cycles, Euler product, functional equations, amenable graphs, approximation by finite graphs.The first and second authors were partially supported by MIUR, GNAMPA and by the European Network "Quantum Spaces -Noncommutative Geometry" HPRN-CT-2002-00280.The third author was partially supported by the National Science Foundation, the Academic Senate of the University of California, and GNAMPA. introductions of [4,39], Ihara's construction can be rephrased in terms of a regular (i.e. constant number of edges spreading from each vertex) finite graph X, its universal covering Y and the corresponding structure group G = π 1 (X). By the homotopic nature of G, one may equivalently represent G-conjugacy classes in terms of suitably reduced primitive cycles on the graph X. Here, a reduced cycle on X (of length m) is a set {e j , j ∈ Z m }, where the starting vertex of e j+1 coincides with the ending vertex of e j , and e j+1 is not the opposite of e j , for j ∈ Z m . Beside...

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Section: Introductionmentioning

confidence: 99%

A trace on fractal graphs and the Ihara zeta function

Guido

Isola

Lapidus

2008

Trans. Amer. Math. Soc.

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“…This theorem is extended to irregular graphs in [Ba], [Ha2], [ST], and [Ho]. The reader is referred to [ST] and the references therein for the history and various zeta functions attached to a graph.…”

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confidence: 99%

The zeta functions of complexes from PGL(3): A representation-theoretic approach

Kang

Wang

2010

Isr. J. Math.

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Abstract. In this paper we obtain a closed form expression of the zeta function Z(X Γ , u) of a finite quotient X Γ of the Bruhat-Tits building of PGL 3 over a nonarchimedean local field F by a discrete cocompact torsion-free subgroup Γ of PGL 3 . Analogous to a graph zeta function, Z(X Γ , u) is a rational function with two different expressions and it satisfies the Riemann hypothesis if and only if X Γ is a Ramanujan complex.

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“…Hashimoto generalized the determinant expression for the Ihara zeta function of an irregular graph by using the adjacency matrix of the oriented line graph (see [4]). …”

Section: Preliminariesmentioning

confidence: 99%

“…In 1989, Hashimoto [4] deduced multi variable zeta functions for bi-regular bipartite graphs. Also, a generalization of the determinant expression of the Ihara zeta function to all finite irregular graphs by its adjacency matrix was performed by Bass [2].…”

Section: Introductionmentioning

confidence: 99%

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The coefficients of the reduced Bartholdi zeta function

Tahaei

Hashemi

2016

Linear Algebra and its Applications

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Abstract. In this paper, we establish a new zeta function based on the Bartholdi zeta function for an undirected graph G called the reduced Bartholdi zeta function. We study the relation between its coefficients and the structure of the graph, and demonstrate that the coefficients count the star subgraphs in the symmetric digraph D(G). Moreover, we investigate the properties of semi principle minors extracted from the adjacency matrix of the oriented line graph of G. We also present a general formula for calculating all the coefficients of the reduced Bartholdi zeta function.

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Artin Type L-Functions and the Density Theorem for Prime Cycles on Finite Graphs

Cited by 51 publications

References 0 publications

A trace on fractal graphs and the Ihara zeta function

A trace on fractal graphs and the Ihara zeta function

The zeta functions of complexes from PGL(3): A representation-theoretic approach

The coefficients of the reduced Bartholdi zeta function

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