# 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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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, $$$$ is a linear subspace and the other conformal classes are the equivalences classes in the quotient $$\mathcal {W}(G)/$$.

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.

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Correspondence to Dmitry Jakobson.