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.

翻译：本文引入并研究了一类受离散曲率启发的新图类。若一个图存在其生成树上的一个分布，使得每个顶点在随机生成树中的期望度不超过二，则我们称该图具有非负电阻性；这类图恰好是允许具有非负电阻曲率的度量存在的图，该离散曲率由Devriendt和Lambiotte引入。我们证明此类图介于哈密顿图与1-坚韧图之间，并且令人惊讶地，一个图具有非负电阻性当且仅当其双倍扩张匹配多面体与其生成树多面体的内部相交。我们进一步研究了非负电阻图的性质与特征，并提出了若干未来研究的问题。