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Conformally covariant operators and conformal invariants on weighted graphs


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.

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  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.


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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.

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Correspondence to Dmitry Jakobson.

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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.

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Jakobson, D., Ng, T., Stevenson, M. et al. Conformally covariant operators and conformal invariants on weighted graphs. Geom Dedicata 174, 339–357 (2015).

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  • 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