Abstract
Abstract The computation of two-point resistances in networks is a classical problem in electric circuit theory and graph theory. Let G be a triangulation graph with n vertices embedded on an orientable surface. Define K(G) to be the graph obtained from G by inserting a new vertex vϕ to each face ϕ of G and adding three new edges (u, vϕ ), (v, vϕ ) and (w, vϕ ), where u, v and w are three vertices on the boundary of ϕ. In this paper, using star-triangle transformation and resistance local-sum rules, explicit relations between resistance distances in K(G) and those in G are obtained. These relations enable us to compute resistance distance between any two points of Kk (G) recursively. As explanation examples, some resistances in several networks are computed, including the modified Apollonian network and networks constructed from tetrahedron, octahedron and icosahedron, respectively.
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