电阻曲率：识别、多面体结构与图乘积
Resistance Curvature: Recognition, Polyhedral Structure, and Graph Products
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中文总结 AI 辅助
本文解决了RN、RP、SRN图的识别复杂度问题，刻画了树双匹配多面体Θ(G)的顶点，确定了k_G，还证明路径有限笛卡尔乘积图为RN图并刻画其RP、SRN子类。
中文摘要 AI 辅助
电阻曲率是由Devriendt和Lambiotte提出的一种基于有效电阻定义的新型离散曲率概念。若存在正边权的选择使得图的每个顶点处的电阻曲率均非负，则称该图为电阻非负图。该性质在随机生成树方面具有显著的组合解释：一个图是电阻非负图当且仅当它能在其生成树上赋予一种分布，使得每个顶点的期望度数至多为2。电阻非负性还可通过树双匹配多面体进行刻画。这些刻画揭示了电阻曲率、有效电阻、生成树分布、匹配理论与多面体组合学之间的紧密联系。基于曲率的符号，Devriendt引入了电阻非负（RN）图、电阻正（RP）图和严格电阻非负（SRN）图这三类图，并提出了关于它们的识别、多面体结构及结构性质的若干问题。在本文中，我们首先回答了Devriendt关于识别RN、RP和SRN图的计算复杂性问题，证明这三类图均可在多项式时间内完成识别。随后，我们解决了他关于树双匹配多面体Θ(G)的问题，还通过满秩紧约束系统刻画了Θ(G)的顶点。当Θ(G)非空时，我们确定了最小正整数k_G，使得k_GΘ(G)是一个格多面体。最后，对于路径的任意有限笛卡尔乘积图，我们显式构造了满足电阻非负条件的平均点，从而证明这类图均为RN图，并进一步刻画了这类笛卡尔乘积图中属于RP或SRN图的子类。
英文摘要
Resistance curvature, introduced by Devriendt and Lambiotte, is a novel discrete curvature notion defined through effective resistance. A graph is called resistance nonnegative if there exists a choice of positive edge weights for which the resistance curvature is nonnegative at every vertex. This property has a notable combinatorial interpretation in terms of random spanning trees: a graph is resistance nonnegative if and only if it admits a distribution on its spanning trees under which every vertex has expected degree at most two. The resistance nonnegativity can also be characterized by the tree double matching polytope. These characterizations reveal strong connections among resistance curvature, effective resistance, spanning tree distributions, matching theory, and polyhedral combinatorics. Based on the sign of the curvature, Devriendt introduced the classes of resistance nonnegative (RN), resistance positive (RP), and strictly resistance nonnegative (SRN) graphs, and posed several questions concerning their recognition, polyhedral structure, and structural properties. In this paper, we first answer Devriendt's question on the computational complexity of recognizing RN, RP, and SRN graphs by proving that all three classes can be recognized in polynomial time. We then address his question concerning the tree double matching polytope $Θ(G)$. Further, we characterize the vertices of $Θ(G)$ in terms of full-rank systems of tight constraints. Whenever $Θ(G)\neq\emptyset$, we also determine the least positive integer $k_G$ such that $k_GΘ(G)$ is a lattice polytope. Finally, for every finite Cartesian product of paths, we explicitly construct an average point satisfying the condition for resistance nonnegativity, thereby obtaining that such graphs are RN. We further characterize the classes of such Cartesian product graphs that are RP or SRN.
