Kirchhoff Graph Uniformity

Abstract

This article discusses Kirchhoff graph uniformity—that all edge vectors in a Kirchhoff graph have the same multiplicity. For a given Kirchhoff graph, an associated digraph is constructed. Based on these graphs, the equivalence of a linear-algebraic condition and a vector graph being Kirchhoff is proven. This condition is then used to show that $2$-connected Kirchhoff graphs are uniform. Other Kirchhoff graphs need not be uniform.