Bartholdi zeta functions of periodic graphs

Sato, Iwao

doi:10.1080/03081080903171914

Linear and Multilinear Algebra

2011

DOI: 10.1080/03081080903171914

|View full text |Cite

Bartholdi zeta functions of periodic graphs

Iwao Sato

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2013

Publication Types

Select...

Other1

Relationship

Self Cite0

Independent1

Authors

Journals

Cited by 1 publication

(1 citation statement)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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%

Zeta Functions for Infinite Graphs and Functional Equations

Guido¹,

Isola²

2013

Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II

View full text Add to dashboard Cite

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.

show abstract

mentioning

confidence: 99%

Zeta Functions for Infinite Graphs and Functional Equations

Guido¹,

Isola²

2013

Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II

View full text Add to dashboard Cite

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

hi@scite.ai

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Bartholdi zeta functions of periodic graphs

Cited by 1 publication

References 16 publications

Zeta Functions for Infinite Graphs and Functional Equations

Zeta Functions for Infinite Graphs and Functional Equations

Contact Info

Product

Resources

About