Skip to main content

Conformally covariant operators and conformal invariants on weighted graphs

Geometriae Dedicata volume 174, pages 339–357 (2015)Cite this article

Abstract

Let \(G\) be a finite connected simple graph. We define the moduli space of conformal structures on \(G\). We propose a definition of conformally covariant operators on graphs, motivated by Graham et al. (J Lond Math Soc 46:557–565, 1992). We provide examples of conformally covariant operators, which include the edge Laplacian and the adjacency matrix on graphs. In the case where such an operator has a nontrivial kernel, we construct conformal invariants, providing discrete counterparts of several results in Canzani et al. (2014; Electron Res Announc Math Sci 20:43–50, 2013) established for Riemannian manifolds. In particular, we show that the nodal sets and nodal domains of null eigenvectors are conformal invariants.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Fig. 1
Fig. 2
Fig. 3

Notes

  1. A bipartite component is a connected component that is also bipartite. Equivalently, \(\omega _0\) is the number of connected components of \(G\) that do not contain an odd cycle.

  2. By an abuse of notation, we consider \(\mathcal {W}(G)\) to be an \(\mathbb {R}\)-vector space of dimension \(|E|\) by identifying a weight \(w = (w_{e})_{e \in E}\) with the vector \((\ln w_{e})_{e \in E} \in \mathbb {R}^{|E|}\). In this way, \([1]\) is a linear subspace and the other conformal classes are the equivalences classes in the quotient \(\mathcal {W}(G)/[1]\).

  3. The manner in which these differential operators transform under a conformal change of weight is analogous to how the GMJS operators transform under a conformal change of Riemannian metric in [9, 10].

  4. Note that \(X_w\) is not dependent on the choice of basis.

References

  1. Bacher, R., de la Harpe, P., Nagnibeda, T.: The lattice of integral flows and the lattice of integral cuts on a finite graph. Bull. Soc. Math. Fr. 125(2), 167–198 (1997)

    MATH  Google Scholar 

  2. Baker, M., Faber, X.: Metrized graphs, Laplacian operators, and electrical networks. Quantum graphs and their applications. Contemp. Math. 415, 15–33 (2006)

  3. Baker, M., Norine, S.: Riemann–Roch and Abel–Jacobi theory on a finite graph. Adv. Math. 215(2), 766–788 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bién, A.: On the determinant of Hexagonal Grids \(H_{k, n}\). arXiv:1309.0087v2

  5. Biggs, N.: Algebraic Graph Theory, 2nd edn. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1993)

    Google Scholar 

  6. Bobenko, A., Pinkall, U., Matthew, B.: Discrete conformal maps and ideal hyperbolic polyhedra. arXiv:1005.2698

  7. Borovicanin, B., Gutman, I.: Nullity of graphs: an updated survey. Zb. Rad. (Beogr.) 14(22), 137–154 (2011). Selected topics on applications of graph spectra

    MathSciNet  Google Scholar 

  8. Branson, T., Gover, R.: The conformal deformation detour complex for the obstruction tensor. Proc. AMS 135(9), 2961–2965 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Canzani, Y., Gover, R., Jakobson, R., Ponge, R.: Conformal Invariants from Nodal Sets. I. Negative Eigenvalues and Curvature Prescription. Int Math Res Notices. 2014, 2356–2400 (2014)

  10. Canzani, Y., Gover, R., Jakobson, D., Ponge, R.: Nullspaces of conformally invariant operators. Applications to \(Q_k\)-curvature. Electron. Res. Announc. Math. Sci. 20, 43–50 (2013)

    MATH  MathSciNet  Google Scholar 

  11. Caracciolo, S., De Grandi, C., Sportiello, A.: Renormalization flow for unrooted forests on a triangular lattice. Nucl. Phys. B 787(3), 260–282 (2007)

    Article  MATH  Google Scholar 

  12. Champion, D., Marchese, A., Miller, J., Young, A.: Constant scalar curvature metrics on boundary complexes of cyclic polytopes. arXiv:1009.3061

  13. Chung, F.R.K.: Spectral graph theory. CBMS Regional Regional Conference Series in Mathematics, vol. 92. AMS, Providence (1997)

  14. Colin de Verdière, Y.: Spectre d’opérateurs différentiels sur les graphes. Random walks and discrete potential theory (Cortona, 1997). Symposium Mathematics, XXXIX, pp. 139–164. Cambridge Univ. Press, Cambridge (1999)

  15. Costello, K., Vu, V.: The rank of random graphs. Random Struct. Algorithms 33(3), 269–285 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dodziuk, J., Patodi, V.K.: Riemannian structures and triangulations of manifolds. J. Indian Math. Soc. (N.S.) 40(1–4), 1–52 (1976). 1977

    MATH  MathSciNet  Google Scholar 

  17. Fallat, S., Hogben, L.: The minimum rank of symmetric matrices described by a graph: a survey. Linear Algebra Appl. 426, 558–582 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Fefferman, C., Graham, C.R.: Conformal invariants. É;ie Cartan et les Mathématiques d’Aujourd’hui, Astérisque, hors série, 95–116 (1985)

  19. Fernandez, L.: The rank of symmetric random matrices via a graph process. M. Sc. Thesis, McGill University (2012)

  20. Glickenstein, D.: Discrete conformal variations and scalar curvature on piecewise flat two- and three-dimensional manifolds. J. Differ. Geom. 87(2), 201–237 (2011)

    MATH  MathSciNet  Google Scholar 

  21. Graham, C.R.: Conformally invariant powers of the Laplacian. II. Nonexistence. J. Lond. Math. Soc. (2) 46(3), 566–576 (1992)

    Article  MATH  Google Scholar 

  22. Graham, C.R., Jenne, R., Mason, L.J., Sparling, G.A.: Conformally invariant powers of the Laplacian, I: existence. J. Lond. Math. Soc. 46, 557–565 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  23. Graham, R.L., Pollak, H.O.: On the addressing problem for loop switching. Bell Syst. Tech. J. 1, 2495–2519 (1971)

    Article  MathSciNet  Google Scholar 

  24. Grossman, J., Kulkarni, D., Devadatta, M., Schochetman. I.: Algebraic graph theory without orientation. In: Proceedings of the 3rd ILAS Conference (Pensacola, FL, 1993). Linear Algebra Appl. 212/213, pp. 289–307 (1994)

  25. Haemers, W., Peeters, M.: The maximum order of adjacency matrices of graphs with a given rank. Des. Codes Cryptogr. 65, 223–232 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kotlov, A., Lovasz, L.: The rank and size of graphs. J. Graph Theory 23, 185–189 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kung, J.: Preface: old and new perspectives on the Tutte polynomial. Ann. Comb. 12, 133–137 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. Luo, F.: Combinatorial Yamabe flow on surfaces. Commun. Contemp. Math. 6(5), 765–780 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. Paneitz, S.: A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (preprint, 1983). Reproduced as: SIGMA 4, 036, 3 pp. (2008)

  30. Pollicott, M., Sharp, R.: A Weil-Petersson type metric on spaces of metric graphs. Geometria Dedicata 172(1), 229–244

  31. Royle, G.: The rank of a cograph. Electron. J. Comb. 10(11) (2003), Note 11

  32. Wilson, S.: Conformal cochains. Trans. AMS 360(10), 5247–5264 (2008)

    Article  MATH  Google Scholar 

  33. Wünsch, V.: On conformally invariant differential operators. Math. Nachr. 129, 269–281 (1986)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors want to have Y. Canzani, R. Choksi, D. Futer, J.-C. Nave, S. Norin, A. Oberman, R. Ponge, I. Rivin and G. Tsogtgerel for useful discussions related to the subject of this paper. The authors want to thank the anonymous referee for useful remarks and corrections.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Str. West, Montreal, QC, H3A 0B9, Canada

    Dmitry Jakobson, Thomas Ng, Matthew Stevenson & Mashbat Suzuki

Authors
  1. Dmitry Jakobson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas Ng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Matthew Stevenson
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mashbat Suzuki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dmitry Jakobson.

Additional information

D. J. was supported by NSERC and FQRNT. T. Ng was supported by ISM and CRM. M. St. was supported by NSERC. M. Suz. was supported by McGill Faculty of Science.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jakobson, D., Ng, T., Stevenson, M. et al. Conformally covariant operators and conformal invariants on weighted graphs. Geom Dedicata 174, 339–357 (2015). https://doi.org/10.1007/s10711-014-0021-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-014-0021-0

Keywords

  • Weighted graph
  • Conformal structure
  • Moduli space
  • Conformally covariant operator
  • Conformal invariant
  • Adjacency matrix
  • Incidence matrix
  • Edge Laplacian
  • Kernel
  • Signature
  • Nodal set

Mathematics Subject Classification (2000)

  • Primary: 05C22
  • Secondary: 05C50
  • 53A30
  • 53A55
  • 58D27
  • 58J50