Applications of Generalized Hierarchical Product of Graphs in Computing the Szeged Index of Chemical Graphs

Let e be an edge of a graph G with endpoints u and v. Define two sets G(1) (e) and G(2)(e), G(1)(e) is the set of vertices closer to u than to v while G(2)(e) consists of those vertices that are closer to v. Then the Szeged index of G is defined as Sz(G) = Sigma(e is an element of E(G))(vertical bar G(1)(e)parallel to G(2)(e)vertical bar). The generalized hierarchical product of graphs was defined by Barriere et. al. (L. Barriere, C. Dalfo, M. A. Fiol and M. Mitjana, Discrete Math., 309 (2009) 3871-3881.) In this paper we compute the Szeged index of hierarchical product of graphs. As a consequence of our results, these indices were computed for some chemical structures such as regular dicentric dendrimers, linear phenylenes, chimer fullerene C-60, truncated cube and truncated cuboctahedron.