On Steinerberger Curvature and Graph Distance Matrices

TL;DR

This paper explores Steinerberger's graph curvature concept, showing its near invariance under certain operations, characterizing distance matrices, and analyzing solutions to related linear systems.

Contribution

It introduces new characterizations of graph distance matrices and their null spaces, and analyzes how graph operations affect Steinerberger curvature.

Findings

Nonnegative curvature is nearly preserved under three graph operations.

Characterization of the distance matrix and its null space after adding edges.

Construction of graphs where the linear system involving the distance matrix has no solution.

Abstract

Steinerberger proposed a notion of curvature on graphs involving the graph distance matrix (J. Graph Theory, 2023). We show that nonnegative curvature is almost preserved under three graph operations. We characterize the distance matrix and its null space after adding an edge between two graphs. Let $D$ be the graph distance matrix and $\mathbf{1}$ be the all-one vector. We provide a way to construct graphs so that the linear system $Dx = \mathbf{1}$ does not have a solution.