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 [25]. 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 [11,12] established for Riemannian manifolds. In particular, we show that the nodal sets and nodal domains of null eigenvectors are conformal invariants. 
Dmitry Jakobson, Thomas Ng, Matthew Stevenson and Mashbat Suzuki
math.CO math.DG math.SP