Generalized Resistance Geometry from Kron Reduction and Effective Resistance

Abstract

We develop a generalized resistance geometry for unsigned directed graphs and signed undirected graphs based on Kron reduction and effective resistance. For unsigned strongly connected weight-balanced directed graphs, we establish a generalized Fiedler--Bapat identity involving an associated signed undirected Laplacian constructed from the symmetrized pseudoinverse. We identify this Laplacian as the sum of the symmetrized directed Laplacian and a positive-semidefinite correction induced by graph asymmetry. This decomposition yields an effective-resistance comparison that extends a previous normality-based result, and we prove that the construction commutes with Kron reduction. For general unsigned strongly connected directed graphs satisfying a positivity condition, we define resistance curvature and resistance radius through a weight-balanced representation. Motivated by the associated signed undirected graphs, we introduce a class whose effective resistances form a metric and show that the resulting resistance matrices are precisely the strict negative type metric matrices. Within this framework, we characterize the unique solution of the maximum graph-variance problem and develop generalized resistive embeddings into Euclidean space.