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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 9 (4 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
A C ∗ algebra of geometric operators on selfsimilar CWcomplexes. Novikov–Shubin and L 2 Betti numbers, preprint
, 2006
"... Abstract. A class of CWcomplexes, called selfsimilar complexes, is introduced, together with C ∗algebras Aj of operators, endowed with a finite trace, acting on squaresummable cellular jchains. Since the Laplacian ∆j belongs to Aj, L 2Betti numbers and NovikovShubin numbers are defined for su ..."
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Cited by 2 (2 self)
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Abstract. A class of CWcomplexes, called selfsimilar complexes, is introduced, together with C ∗algebras Aj of operators, endowed with a finite trace, acting on squaresummable cellular jchains. Since the Laplacian ∆j belongs to Aj, L 2Betti numbers and NovikovShubin numbers are defined for such complexes in terms of the trace. In particular a relation involving the EulerPoincaré characteristic is proved. L 2Betti and NovikovShubin numbers are computed for some selfsimilar complexes arising from selfsimilar fractals. 1. Introduction. In this paper we address the question of the possibility of extending the definition of some L 2invariants, like the L 2Betti numbers and NovikovShubin numbers, to geometric structures which are not coverings of compact spaces. The first attempt in this sense is due to John Roe [29], who defined a trace
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
Obtaining Upper . . . From Lower Bounds
, 2007
"... We show that a neardiagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an ondiagonal upper bound. If in addition the Dirichlet form is local and regular, then we obtain a full offdiagonal upper bound of the heat kernel provided the ..."
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We show that a neardiagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an ondiagonal upper bound. If in addition the Dirichlet form is local and regular, then we obtain a full offdiagonal upper bound of the heat kernel provided the Dirichlet heat kernel on any ball satisfies a neardiagonal lower estimate. This reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel.
Motif based hierarchical random graphs: structural properties and critical points of an Ising model ∗
, 2010
"... Rev. E, 2006, 73, 066126], according to which the shortrange bonds are nonrandom, whereas the longrange bonds appear independently with the same probability. A number of structural properties of the graphs have been described, among which there are degree distributions, clustering, amenability, s ..."
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Rev. E, 2006, 73, 066126], according to which the shortrange bonds are nonrandom, whereas the longrange bonds appear independently with the same probability. A number of structural properties of the graphs have been described, among which there are degree distributions, clustering, amenability, smallworld property. For one of the motifs, the critical point of the Ising model defined on the corresponding graph has been studied.