Abstract

This paper shows how to extract permutation invariant graph characteristics from the Ihara zeta function. In a previous paper, we have shown that the Ihara zeta function leads to a polynomial characterization of graph structure, and we have shown empirically that the coefficients of the polynomial can be used as to cluster graphs. The aim in this paper is to take this study further by showing how to select the most significant coefficients and how these can be used to gauge graph similarity. Experiments on real-world datasets reveal that the selected coefficients give results that are significantly better than those obtained with the Laplacian spectrum.

Keywords

Zeta Function Cluster Performance Edit Distance Feature Distance Rand Index 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Bai, X., Hancock, E.R.: Recent results on heat kernel embedding of graphs. In: Brun, L., Vento, M. (eds.) GbRPR 2005. LNCS, vol. 3434, pp. 373–382. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  2. 2.
    Bass, H.: The Ihara-Selberg zeta function of a tree lattice. International Journal of Mathematics 6(3), 717–797 (1992)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brooks, B.P.: The coefficients of the characteristic polynomial in terms of the eigenvalues and the elements of an n×n matrix. Applied Mathematics letters 19(6), 511–515 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chung, F.K.: Spectral graph theory. American Mathematical Society (1997)Google Scholar
  5. 5.
    Haemers, W.H., Spence, E.: Enumeration of cospectral graphs. European Journal of Combinatorics 25(2), 199–211 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Hashimoto, K.: Zeta functions of finite graphs and representations of p-adic groups. Advanced Study of Pure Mathematics 15, 211–280 (1989)MathSciNetGoogle Scholar
  7. 7.
    Ihara, Y.: Discrete subgroups of PL(2, k ϕ). In: Proceeding Symposium of Pure Mathematics, pp. 272–278 (1965)Google Scholar
  8. 8.
    Ihara, Y.: On discrete subgroups of the two by two projective linear group over p-adic fields. Journal of Mathematics Society Japan 18, 219–235 (1996)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kotani, M., Sunada, T.: Zeta functions of finite graphs. Journal of Mathematics University of Tokyo 7(1), 7–25 (2000)MathSciNetMATHGoogle Scholar
  10. 10.
    Luo, B., Wilson, R.C., Hancock, E.R.: Spectral embedding of graphs. Pattern Recognition 36(10), 2213–2223 (2003)CrossRefMATHGoogle Scholar
  11. 11.
    Rosenberg, S.: The Laplacian on a Riemannian manifold. Cambridge University Press, Cambridge (2002)Google Scholar
  12. 12.
    Ren, P., Wilson, R.C., Hancock, E.R.: Pattern vectors from the Ihara zeta function. In: IEEE International Conference on Pattern Recognition (submitted, 2008)Google Scholar
  13. 13.
    Sanfeliu, A., Fu, K.S.: A distance measure between attributed relational graphs for pattern recognition. IEEE Transanctions on Systems, Man, and Cybernetics 13, 353–362 (1983)CrossRefMATHGoogle Scholar
  14. 14.
    Scott, G., Storm, C.K.: The coefficients of the Ihara zeta function. Involve - A Journal of Mathematics 1(2), 217–233 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Storm, C.K.: The zeta function of a hypergraph. Electronic Journal of Combinatorics 13(1) (2006)Google Scholar
  16. 16.
    Wilson, R.C., Luo, B., Hancock, E.R.: Pattern vectors from algebraic graph theory. IEEE Transactions on Pattern Analysis and Machine Intelligence 27(7), 1112–1124 (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Peng Ren
    • 1
  • Richard C. Wilson
    • 1
  • Edwin R. Hancock
    • 1
  1. 1.Department of Computer ScienceThe University of YorkYorkUK

Personalised recommendations