Ihara Zeta Functions for Periodic Simple Graphs
The definition and main properties of the Ihara zeta function for graphs are reviewed, focusing mainly on the case of periodic simple graphs. Moreover, we give a new proof of the associated determinant formula, based on the treatment developed by Stark and Terras for finite graphs.
KeywordsPeriodic graphs Ihara zeta function analytic determinant determinant formula functional equations
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- H. Bass, A. Lubotzky. Tree lattices, Progress in Math. 176, Birkhäuser, Boston, 2001.Google Scholar
- B. Clair. Zeta functions of graphs with Z actions, preprint, 2006, arXiv:math.NT/0607689.Google Scholar
- R.I. Grigorchuk, A. Żuk. The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps, in: “Random Walks and Geometry”, Proc. Workshop (Vienna, 2001), V.A. Kaimanovich et al., eds., de Gruyter, Berlin, 2004, pp. 141–180.Google Scholar
- D. Guido, T. Isola, M.L. Lapidus. A trace on fractal graphs and the Ihara zeta function, to appear in Trans. Amer. Math. Soc., arXiv:math.OA/0608060.Google Scholar
- D. Guido, T. Isola, M.L. Lapidus. Ihara’s zeta function for periodic graphs and its approximation in the amenable case, preprint, 2006, arXiv:math.OA/0608229.Google Scholar
- K. Hashimoto, A. Hori. Selberg-Ihara’s zeta function for p-adic discrete groups, in: “Automorphic Forms and Geometry of Arithmetic Varieties”, Adv. Stud. Pure Math. 15, Academic Press, Boston, MA, 1989, pp. 171–210.Google Scholar
- K. Hashimoto. Zeta functions of finite graphs and representations of p-adic groups, in: “Automorphic Forms and Geometry of Arithmetic Varieties”, Adv. Stud. Pure Math. 15, Academic Press, Boston, MA, 1989, pp. 211–280.Google Scholar
- M.D. Horton, H.M. Stark, A.A. Terras. What are zeta functions of graphs and what are they good for?, Quantum graphs and their applications, 173–189, Contemp. Math., 415, Amer. Math. Soc., Providence, RI, 2006.Google Scholar
- A. Lubotzky. Discrete groups, expanding graphs and invariant measures, Progress in Math. 125, Birkhäuser, Basel, 1994.Google Scholar