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

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