Resistance distances in extended or contracted networks

Abstract

In this paper we introduce new effective resistances on a given network, associated with a positive value and two weights on the vertex set and investigate under which conditions they determine a distance. We prove that this property is closely related with superharmonicity. Moreover, we analyze the behavior of these distances under the usual network transformations, specially the so-called star-mesh transformation. We also compute the effective resistance for an extended network; that is, the network obtained from the former one by joining a new vertex, and then study the effect of the contraction of this new network; that is, we apply a star-mesh transformation with center in the joined vertex.