Bartholdi zeta functions for periodic simple graphs

Guido, Daniele; Isola, Tommaso; Lapidus, Michel L.

doi:10.1090/pspum/077/2459866

Cited by 4 publications

(7 citation statements)

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“…Proof. We consider only the case of self-similar graphs, for the periodic case see [12]. Denote by (C, v) the closed path C with the origin in v ∈ V X.…”

Section: Definition 14 (Finite Propagation Operators) a Bounded Linmentioning

confidence: 99%

“…Given C, D ∈ K, we say that C and D are G-equivalent, and write C ∼ G D, if there is a local isomorphism γ ∈ G such that D = γ(C). We denote by [K] G the set of G-equivalence classes of cycles, and analogously for the subset P. The notion of Γ-equivalence is analogous (see [12] for details), and we denote by [·] G also a Γ-equivalence class.…”

Section: Definition 14 (Finite Propagation Operators) a Bounded Linmentioning

confidence: 99%

“…(i) Let C ∈ K, then the following limit exists and is finite: Proof. We prove only the self-similar case, for the periodic case see [12]. We have successively:…”

Section: Definition 14 (Finite Propagation Operators) a Bounded Linmentioning

confidence: 99%

“…We quote [5,18,19,20] for further results and generalizations of the Bartholdi zeta function. The extension to the case of infinite periodic simple graphs is contained in [12], where a functional equation for regular graphs and a determinant formula are proved. Sato [24] generalised the determinant formula to the non simple case, and also proved it for the case of fractal graphs [23].…”

mentioning

confidence: 99%

“…The aim of this paper is two-fold: on the one hand we illustrate all the mentioned results both for the periodic and the fractal case, using a unified approach in all the statements and also in some proofs, while for others we only treat the fractal case, referring the readers to [12] for the periodic case. On the other hand, we analyze the meaning and validity of the functional equation for infinite graphs.…”

mentioning

confidence: 99%

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Zeta Functions for Infinite Graphs and Functional Equations

Guido¹,

Isola²

2013

Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II

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Abstract. The definitions and main properties of the Ihara and Bartholdi zeta functions for infinite graphs are reviewed. The general question of the validity of a functional equation is discussed, and various possible solutions are proposed. IntroductionIn this paper, we review the main results concerning the Ihara zeta function and the Bartholdi zeta function for infinite graphs. Moreover, we propose various possible solutions to the problem of the validity of a functional equation for those zeta functions.The zeta function associated to a finite graph by Ihara, Sunada, Hashimoto and others, combines features of Riemann's zeta function, Artin L-functions, and Selberg's zeta function, and may be viewed as an analogue of the Dedekind zeta function of a number field [3,14,15,16,17,25,26]. It is defined by an Euler product over proper primitive cycles of the graph.A main result for the Ihara zeta function Z X (z) associated with a graph X, is the so called determinant formula, which shows that the inverse of this function can be written, up to a polynomial, as det(I − Az + Qz 2 ), where A is the adjacency matrix and Q is the diagonal matrix corresponding to the degree minus 1. As a consequence, for a finite graph, Z X (z) is indeed the inverse of a polynomial, hence can be extended meromorphically to the whole plane.A second main result is the fact that, for (q + 1)-regular graphs, namely graphs with degree constantly equal to (q + 1), Z X , or better its so called completion ξ X , satisfies a functional equation, namely is invariant under the transformation z → 1 qz . The first of the mentioned results has been proved for infinite (periodic or fractal) graphs in [11,10], by introducing the analytic determinant for operator 2000 Mathematics Subject Classification. 05C25; 05C38; 46Lxx; 11M41.

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“…Proof. We consider only the case of self-similar graphs, for the periodic case see [12]. Denote by (C, v) the closed path C with the origin in v ∈ V X.…”

Section: Definition 14 (Finite Propagation Operators) a Bounded Linmentioning

confidence: 99%

Section: Definition 14 (Finite Propagation Operators) a Bounded Linmentioning

confidence: 99%

“…(i) Let C ∈ K, then the following limit exists and is finite: Proof. We prove only the self-similar case, for the periodic case see [12]. We have successively:…”

Section: Definition 14 (Finite Propagation Operators) a Bounded Linmentioning

confidence: 99%