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A trace on fractal graphs AND THE IHARA ZETA FUNCTION
, 2008
"... 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 MokhtariSharghi have studied zeta functions for infinite graphs acted u ..."
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Cited by 13 (5 self)
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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 MokhtariSharghi have 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. The Ihara zeta function, originally associated to certain groups and then combinatorially
IHARA’S ZETA FUNCTION FOR PERIODIC GRAPHS AND ITS APPROXIMATION IN THE AMENABLE CASE
, 2008
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ZETA FUNCTIONS OF GRAPHS WITH Z ACTIONS
, 2006
"... Abstract. Suppose Y is a regular covering of a 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 functio ..."
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Cited by 4 (1 self)
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Abstract. Suppose Y is a regular covering of a 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. 1.
Bartholdi zeta functions for periodic simple graphs
, 2008
"... The definition of the Bartholdi zeta function is extended to the case of infinite periodic graphs. By means of the analytic determinant for semifinite von Neumann algebras studied by the authors in [7], a determinant formula and functional equations are obtained for this zeta function. ..."
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Cited by 2 (1 self)
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The definition of the Bartholdi zeta function is extended to the case of infinite periodic graphs. By means of the analytic determinant for semifinite von Neumann algebras studied by the authors in [7], a determinant formula and functional equations are obtained for this zeta function.
Bartholdi Zeta Functions of Fractal Graphs
"... Recently, Guido, Isola and Lapidus [11] defined the Ihara zeta function of a fractal graph, and gave a determinant expression of it. We define the Bartholdi zeta function of a fractal graph, and present its determinant expression. 1 ..."
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Recently, Guido, Isola and Lapidus [11] defined the Ihara zeta function of a fractal graph, and gave a determinant expression of it. We define the Bartholdi zeta function of a fractal graph, and present its determinant expression. 1
ZETA FUNCTIONS OF INFINITE GRAPH BUNDLES
, 2007
"... Abstract. We compute the equivariant zeta function for bundles over infinite graphs and for infinite covers. In particular, we give a “transfer formula ” for the zeta function of infinite graph covers. Also, when the infinite cover is given as a limit of finite covers, we give a formula for the limi ..."
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Abstract. We compute the equivariant zeta function for bundles over infinite graphs and for infinite covers. In particular, we give a “transfer formula ” for the zeta function of infinite graph covers. Also, when the infinite cover is given as a limit of finite covers, we give a formula for the limit of the zeta functions. 1.
THE IHARA ZETA FUNCTION OF THE INFINITE GRID
"... Abstract. The infinite grid is the Cayley graph of Z × Z with the usual generators. In this paper, the Ihara zeta function for the infinite grid is computed using elliptic integrals and theta functions. The zeta function of the grid extends to an analytic, multivalued function which satisfies a fun ..."
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Abstract. The infinite grid is the Cayley graph of Z × Z with the usual generators. In this paper, the Ihara zeta function for the infinite grid is computed using elliptic integrals and theta functions. The zeta function of the grid extends to an analytic, multivalued function which satisfies a functional equation. The set of singularities in its domain is finite. The grid zeta function is the first computed example which is nonelementary, and which takes infinitely many values at each point of its domain. It is also the limiting value of the normalized sequence of Ihara zeta functions for square grid graphs and torus graphs. 1.