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Ihara Zeta Functions for Periodic Simple Graphs

  • Daniele Guido
  • Tommaso Isola
  • Michel L. Lapidus
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

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.

Keywords

Periodic graphs Ihara zeta function analytic determinant determinant formula functional equations 

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References

  1. [1]
    L. Bartholdi. Counting paths in graphs, Enseign. Math. 45 (1999), 83–131.zbMATHMathSciNetGoogle Scholar
  2. [2]
    H. Bass. The Ihara-Selberg zeta function of a tree lattice, Internat. J. Math. 3 (1992), 717–797.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    H. Bass, A. Lubotzky. Tree lattices, Progress in Math. 176, Birkhäuser, Boston, 2001.Google Scholar
  4. [4]
    B. Clair, S. Mokhtari-Sharghi. Zeta functions of discrete groups acting on trees, J. Algebra 237 (2001), 591–620.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    B. Clair, S. Mokhtari-Sharghi. Convergence of zeta functions of graphs, Proc. Amer. Math. Soc. 130 (2002), 1881–1886.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    B. Clair. Zeta functions of graphs with Z actions, preprint, 2006, arXiv:math.NT/0607689.Google Scholar
  7. [7]
    D. Foata, D. Zeilberger. A combinatorial proof of Bass’s evaluations of the Ihara-Selberg zeta function for graphs, Trans. Amer. Math. Soc. 351 (1999), 2257–2274.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    B. Fuglede, R.V. Kadison. Determinant theory in finite factors, Ann. Math. 55 (1952), 520–530.CrossRefMathSciNetGoogle Scholar
  9. [9]
    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
  10. [10]
    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
  11. [11]
    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
  12. [12]
    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
  13. [13]
    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
  14. [14]
    K. Hashimoto. On zeta and L-functions of finite graphs, Internat. J. Math. 1 (1990), 381–396.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    K. Hashimoto. Artin type L-functions and the density theorem for prime cycles on finite graphs, Internat. J. Math. 3 (1992), 809–826.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    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
  17. [17]
    Y. Ihara. On discrete subgroups of the two by two projective linear group over \( \mathfrak{p} \)-adic fields, J. Math. Soc. Japan 18 (1966), 219–235.zbMATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    M. Kotani, T. Sunada. Zeta functions of finite graphs, J. Math. Sci. Univ. Tokyo 7 (2000), 7–25.zbMATHMathSciNetGoogle Scholar
  19. [19]
    A. Lubotzky. Discrete groups, expanding graphs and invariant measures, Progress in Math. 125, Birkhäuser, Basel, 1994.Google Scholar
  20. [20]
    H. Mizuno, I. Sato. Bartholdi zeta functions of some graphs, Discrete Math. 206 (2006), 220–230.CrossRefMathSciNetGoogle Scholar
  21. [21]
    B. Mohar. The spectrum of an infinite graph, Linear Algebra Appl. 48 (1982), 245–256.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    B. Mohar, W. Woess. A survey on spectra of infinite graphs, Bull. London Math. Soc. 21 (1989), 209–234.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    S. Northshield. A note on the zeta function of a graph, J. Combin. Theory Series B 74 (1998), 408–410.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    J.-P. Serre. Trees, Springer-Verlag, New York, 1980.zbMATHGoogle Scholar
  25. [25]
    J.-P. Serre. Répartition asymptotique des valeurs propres de l’opérateur de Hecke Tp, J. Amer. Math. Soc. 10 (1997), 75–102.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    H.M. Stark, A.A. Terras. Zeta functions of finite graphs and coverings, Adv. Math. 121 (1996), 126–165.CrossRefMathSciNetGoogle Scholar
  27. [27]
    H.M. Stark, A.A. Terras. Zeta functions of finite graphs and coverings. II, Adv. Math. 154 (2000), 132–195.zbMATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    H.M. Stark, A.A. Terras. Zeta functions of finite graphs and coverings. III, Adv. Math. 208 (2007), 467–489.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    T. Sunada. L-functions in geometry and applications, Springer Lecture Notes in Math. 1201, 1986, pp. 266–284.MathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Daniele Guido
    • 1
  • Tommaso Isola
    • 1
  • Michel L. Lapidus
    • 2
  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly
  2. 2.Department of MathematicsUniversity of CaliforniaRiversideUSA

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