A measure-theoretic representation of graphs

TL;DR

This paper introduces a measure-theoretic framework for representing graphs and matrices, providing a new metric for graph isomorphism classes and simplifying the understanding of action convergence in graph limit theory.

Contribution

It develops a measure-theoretic representation of matrices that defines a new pseudo-metric, which becomes a true metric on adjacency and Laplacian matrices, aiding graph analysis.

Findings

Defines a measure-theoretic representation of matrices.

Establishes a metric for graph isomorphism classes.

Simplifies the understanding of action convergence.

Abstract

Inspired by the notion of action convergence in graph limit theory, we introduce a measure-theoretic representation of matrices, and we use it to define a new notion of pseudo-metric on the space of matrices. Moreover, we show that such pseudo-metric is a metric on the subspace of adjacency or Laplacian matrices for graphs. Hence, in particular, we obtain a metric for isomorphism classes of graphs. Additionally, we study how some properties of graphs translate in this measure representation, and we show how our analysis contributes to a simpler understanding of action convergence of graphops.