Computation of Topological Indices of Some Graphs

Let G=(V,E) be a simple connected graph with vertex set V and edge set E. The Wiener index of G is defined by W(G)=∑{x,y}⊆Vd(x,y), where d(x,y) is the length of the shortest path from x to y. The Szeged index of G is defined by Sz(G)=∑e=uv∈Enu(e|G)nv(e|G), where nu(e|G) (resp. nv(e|G)) is the number of vertices of G closer to u (resp. v) than v (resp. u). The Padmakar–Ivan index of G is defined by PI(G)=∑e=uv∈E[neu(e|G)+nev(e|G)], where neu(e|G) (resp. nev(e|G)) is the number of edges of G closer to u (resp. v) than v (resp. u). In this paper we find the above indices for various graphs using the group of automorphisms of G. This is an efficient method of finding these indices especially when the automorphism group of G has a few orbits on V or E. We also find the Wiener indices of a few graphs which frequently arise in mathematical chemistry using inductive methods.