Some rules on resistance distance with applications

In this work, some rules for resistance distances of a graph G are established. Let S be a set of vertices of G such that all vertices in S have the same neighborhood N in G − S. If |S| = 2, 3, 4, simple formulae are derived to compute resistance distances between vertices in S in terms of the cardinality of N. These show that resistance distances between vertices in S depend only on the cardinality of N and the induced subgraph G[S]. One question arises naturally: does this property hold for S with arbitrarily many vertices? We answer this question by the following reduction principle: resistance distances between vertices in S can be computed as in the subgraph obtained from G[S N] by deleting all the edges between vertices in N.

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