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This article discusses a geometric perspective on the well-known fact ingraph theory that the effective resistance is a metric on the nodes of a graph.The classical proofs of this fact make use of ideas from electrical circuits orrandom walks; here we describe an alternative approach which combines geometric(using simplices) and algebraic (using the Schur complement) ideas. Theseperspectives are unified in a matrix identity of Miroslav Fiedler, whichbeautifully summarizes a number of related ideas at the intersection of graphs,Laplacian matrices and simplices, with the metric property of the effectiveresistance as a prominent consequence.

Effective resistance is more than distance: Laplacians, Simplices and the Schur complement