Resistance Distances Of Three Iterative Networks And Its Application

Given a connected graph G=?V?G?,E?G??and a non-negative weight mapping w:E?G??R⁺,we can view?G,w?as an electrical network with w?e?as the conductance of edge e?i.e.,1/w?e?is the resistance of e?.The effective resistance between any two points in networks is a basic problem in electric theory.When w?e?=1 for all the edges in G,then the effective resistance between any two points in G is called the resistance distance between them which is not only a distance function or?intrinsic?metric on graphs but also is the isomorphism invariant on graphs.Let G be a triangulation graph with n vertices embedded on an orientable surface.we can obtain vertex-face networks of G by repeatedly inserting a new vertex v_?to each face?of G and adding three new edges?u,v_??,?v,v_??and?w,v_??,where u,v and w are three vertices on the boundary of?.Similar to the construction of vertex-face networks,an Apollonian network is formed by a process of recursively subdividing a triangle into three small triangles.Self-similar?x,y?-flowers networks start from an edge and constantly replaces each exiting edge by two paths of length x and y.In this paper,by using series and parallel principles,star-triangle transformation and partial sum rules,we obtain recursive formulas and matrix expressions of resistance distances in three iterative networks described above.Based on these results,we get resistance distances in the vertex-face networks of the complete graph K₃,tetrahedron,octahedron and icosahedron,respectively.We also compute explicit expressions for some resistances in an Apollonian network,?1,3?-flower networks and?2,2?-flower networks.