This report considers the resistance distance as a recently proposed new intrinsic metric on(molecular) graphs, and in particular, the sum R over resistance distances between all pairs of vertices is considered as a graph invariant. It has been proved that R(G_N) > R(K_N), where G_N denotes a connected graph containing N vertices and K_N denotes a complete graph containing N vertices. The formulas to obtain the R for two classes of regular graphs (cycles and complete graphs) are derived. numerical values of R for four Platonic molecules are also given. They ordered the considered Platonic solids as the icosahedron, the cube, the octahedron, and the tetrahedron according to complexity of their Schlegel graphs. This order agrees with those obtained by many other, frequently used descriptors.
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