Graphs with nonnegative resistance curvature

TL;DR

This paper introduces resistance nonnegative graphs, a new class characterized by a discrete curvature related to spanning trees and metric properties, bridging Hamiltonian and 1-tough graphs.

Contribution

It defines resistance nonnegative graphs, explores their properties, and establishes their relation to spanning tree and matching polytopes, advancing discrete curvature theory.

Findings

Resistance nonnegative graphs lie between Hamiltonian and 1-tough graphs.

A graph is resistance nonnegative iff its twice-dilated matching polytope intersects the interior of its spanning tree polytope.

The paper characterizes and discusses properties of resistance nonnegative graphs.

Abstract

This article introduces and studies a new class of graphs motivated by discrete curvature. We call a graph resistance nonnegative if there exists a distribution on its spanning trees such that every vertex has expected degree at most two in a random spanning tree; these are precisely the graphs that admit a metric with nonnegative resistance curvature, a discrete curvature introduced by Devriendt and Lambiotte. We show that this class of graphs lies between Hamiltonian and $1$-tough graphs and, surprisingly, that a graph is resistance nonnegative if and only if its twice-dilated matching polytope intersects the interior of its spanning tree polytope. We study further characterizations and basic properties of resistance nonnegative graphs and pose several questions for future research.