A Triangle Inequality for p-Resistance

The geodesic distance (path length) and effective resistance are both metrics defined on the vertices of a graph. The effective resistance is a more refined measure of connectivity than the geodesic distance. For example if there are k edge disjoint paths of geodesic distance d between two vertices, then the effective resistance is no more than d k . Thus, the more paths, the closer the vertices. We continue the study of the recently introduced p-effective resistance [9]. The main technical contribution of this note is to prove that the p-effective resistance is a metric for p ∈ (1, 2] and obeys a strong triangle inequality. An easy consequence of this inequality is that we may efficiently find a k-center clustering within a factor of 2p−1 from the optimal clustering with respect to p-effective resistance.