Graphs with nonnegative resistance curvature

2024-10-10

arXiv.org

AI Summary

This paper investigates structural properties of graphs with nonnegative effective resistance curvature—i.e., graphs admitting a spanning tree distribution under which the expected degree of each vertex in a random spanning tree is at most two. We establish, for the first time, an exact equivalence with polyhedral geometry: nonnegative resistance curvature holds if and only if the double-expansion matching polytope intersects the interior of the spanning tree polytope. Combining discrete curvature theory, probabilistic spanning tree modeling, combinatorial polyhedral analysis, and graph-theoretic techniques, we derive a necessary and sufficient geometric characterization. Moreover, we precisely locate this property within the topological hierarchy: it strictly lies between Hamiltonicity and 1-toughness. Our results unify discrete curvature, network robustness, and convex combinatorial structures, offering a novel paradigm for interdisciplinary research at the interface of discrete geometry and complex networks, and raising several open problems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Characterize graphs with nonnegative resistance curvature
• Relate resistance nonnegative graphs to Hamiltonian and 1-tough graphs
• Explore properties of graphs with twice-dilated matching polytope intersections

Innovation

• Graphs with nonnegative resistance curvature
• Distribution on spanning trees
• Intersection of matching and tree polytopes