Graphs with nonnegative resistance curvature
<|reference_start|>Graphs with nonnegative resistance curvature: 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.<|reference_end|>
@article{devriendt2024graphs,
    title={Graphs with nonnegative resistance curvature},
    author={Karel Devriendt},
    journal={arXiv preprint arXiv:2410.07756},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.07756},
    primaryClass={math.CO cs.DM math.MG}
}