Relationships between cycle spaces, gain graphs, graph coverings, fundamental groups, path homology, and graph curvature

TL;DR

This paper establishes a homology vanishing theorem for graphs with positive Bakry-Émery curvature, showing their first homology group is trivial and their fundamental group is finite, linking curvature to topological properties.

Contribution

It introduces new relationships among graph curvature, cycle spaces, gain graphs, and fundamental groups, extending classical manifold results to graph theory.

Findings

Graphs with positive curvature have trivial first homology.

Fundamental groups of such graphs are finite.

New connections between gain graphs, coverings, and homotopy are established.

Abstract

We prove a homology vanishing theorem for graphs with positive Bakry-\'Emery curvature, analogous to a classic result of Bochner on manifolds \cite{Bochner}. Specifically, we prove that if a graph has positive curvature at every vertex, then its first homology group is trivial, where the notion of homology that we use for graphs is the path homology developed by Grigor'yan, Lin, Muranov, and Yau \cite{Grigoryan2}. %\Hm{added the fundamental group curvature relation} We moreover prove that the fundamental group is finite for graphs with positive Bakry-\'Emery curvature, analogous to a classic result of Myers on manifolds \cite{Myers1941}. The proofs draw on several separate areas of graph theory. We study graph coverings, gain graphs, and cycle spaces of graphs, in addition to the Bakry-\'Emery curvature and the path homology. The main results follow as a consequence of several different relationships developed among these different areas. Specifically, we show that a graph with positive curvature can have no non-trivial infinite cover preserving 3-cycles and 4-cycles, and give a combinatorial interpretation of the first path homology in terms of the cycle space of a graph. We relate cycle spaces of graphs to gain graphs with abelian gain group, and relate these to coverings of graphs. Along the way, we prove other new facts about gain graphs, coverings, and cycles spaces that are of related interest. Furthermore, we relate gain graphs to graph homotopy and the fundamental group developed by Grigor'yan, Lin, Muranov, and Yau \cite{Grigoryan_homotopy}, and obtain an alternative proof to their result that the abelianization of the fundamental group is isomorphic to the first path homology over the integers.