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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A trace on fractal graphs and the Ihara zeta function


Authors: Daniele Guido, Tommaso Isola and Michel L. Lapidus
Journal: Trans. Amer. Math. Soc. 361 (2009), 3041-3070
MSC (2000): Primary 11M41, 46Lxx, 05C38; Secondary 05C50, 28A80, 11M36, 30D05
Published electronically: December 29, 2008
MathSciNet review: 2485417
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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 self-similarity 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.


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Additional Information

Daniele Guido
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, I–00133 Roma, Italy
Email: guido@mat.uniroma2.it

Tommaso Isola
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, I–00133 Roma, Italy
Email: isola@mat.uniroma2.it

Michel L. Lapidus
Affiliation: Department of Mathematics, University of California, Riverside, California 92521-0135
Email: lapidus@math.ucr.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04702-8
PII: S 0002-9947(08)04702-8
Keywords: 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
Received by editor(s): May 31, 2007
Published electronically: December 29, 2008
Additional Notes: 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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.