Conformally Covariant Operators and Conformal Invariants on Weighted Graphs

TL;DR

This paper introduces conformally covariant operators on finite graphs, constructs conformal invariants, and explores their properties, providing discrete analogs of classical Riemannian results.

Contribution

It defines conformally covariant operators on graphs and constructs conformal invariants, extending concepts from Riemannian geometry to discrete graph structures.

Findings

Edge Laplacian and adjacency matrix are conformally covariant operators.

Nodal sets and domains of null eigenvectors are conformal invariants.

Provides discrete counterparts of Riemannian conformal invariants.

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 [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.