Remarks on the Spectral Approach to Finding Short Paths

TL;DR

This paper investigates the relationship between spectral properties and shortest paths in graphs, providing counterexamples and positive results under curvature conditions to clarify their connection.

Contribution

It offers new constructions showing spectral properties can diverge from distance functions and introduces curvature-based conditions that align spectral and distance characteristics.

Findings

Spectral properties can contradict shortest path distances in certain graphs.

Curvature conditions can ensure spectral and distance properties are aligned.

Counterexamples demonstrate limitations of spectral methods for shortest path estimation.

Abstract

We are interested in the general question: to what extent are the spectral properties of a graph connected to the distance function? Our motivation is a concrete example of this question that is due to Steinerberger. We provide some negative results via constructions of families of graphs where the spectral properties disagree with the distance function in a manner that answers many of the questions posed by Steinerberger. We also provide a positive result by replacing Steinerberger's set-up with conditions involving graph curvature.