# Conformally covariant operators and conformal invariants on weighted graphs

## 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.

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## 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.

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## 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

### Corresponding author

Correspondence to Dmitry Jakobson.

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

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

### Keywords

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

### Mathematics Subject Classification (2000)

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