On the image of graph distance matrices

TL;DR

This paper investigates the solvability of the linear system involving the graph distance matrix, proving the existence of counterexamples for all graphs with at least 7 vertices and showing that for Erdős-Rényi graphs, the matrix is typically invertible.

Contribution

It provides the first proof of the existence of graphs with non-solvable distance matrix systems for all sizes n ≥ 7 and analyzes the invertibility of these matrices in random graph models.

Findings

Counterexamples exist for all n ≥ 7.

Distance matrices of Erdős-Rényi graphs are invertible with high probability.

Structural properties of the Perron-Frobenius eigenvector are explored.

Abstract

Let $G=(V,E)$ be a finite, simple, connected, combinatorial graph on $n$ vertices and let $D \in \mathbb{R}^{n \times n}$ be its graph distance matrix $D_{ij} = d(v_i, v_j)$. Steinerberger (J. Graph Theory, 2023) empirically observed that the linear system of equations $Dx =\mathbf{1}$, where $\mathbf{1} = (1,1,\dots, 1)^{T}$, very frequently has a solution (even in cases where $D$ is not invertible). The smallest nontrivial example of a graph where the linear system is not solvable are two graphs on 7 vertices. We prove that, in fact, counterexamples exists for all $n\geq 7$. The construction is somewhat delicate and further suggests that such examples are perhaps rare. We also prove that for Erd\H{o}s-R\'enyi random graphs the graph distance matrix $D$ is invertible with high probability. We conclude with some structural results on the Perron-Frobenius eigenvector for a distance matrix.