Distance Exceptional Graphs and the Curvature Index

TL;DR

This paper introduces the curvature index to characterize distance exceptional graphs, providing new insights into their structure, properties, and methods for constructing such graphs, expanding understanding in discrete curvature on graphs.

Contribution

It establishes the curvature index as a key invariant for distance exceptional graphs and develops calculus for this invariant under various graph operations.

Findings

Distance exceptional graphs are characterized by a vanishing curvature index.

Any graph can be embedded as an induced subgraph in a distance exceptional graph.

The paper generalizes known results and offers new methods for constructing distance exceptional graphs.

Abstract

A graph $G=(V,E)$ on $n$ vertices is said to be \emph{distance exceptional} if the equation $D\vec{x} = \vec{1}$ admits no solution $\vec{x}\in\mathbb{R}^{n}$, where $D\in\mathbb{R}^{n\times n}$ is the shortest path distance matrix of $G$. These graphs were first studied by Steinerberger in the context of a notion of discrete curvature (``Curvature on graphs via equilibrium measures,'' \emph{Journal of Graph Theory}, 103(3), 2023). This work has led to several open questions about distance exceptional graphs, including: What is the structure of such graphs? How can they be characterized? How rare are they? In this paper, we investigate these questions through the lens of a graph invariant we term the \emph{curvature index}. We show that a graph is distance exceptional if and only if this invariant vanishes, and we develop a calculus for this invariant under graph operations including the Cartesian product and graph join. As a result, we recover and generalize a number of known results in this area. We show that any graph $G$ can be realized as an induced subgraph of a distance exceptional graph $G'$. Moreover, in many cases, this embedding is an isometry. In turn, this leads to a number of methods for constructing distance exceptional graphs.