Vertex and edge Padmakar-Ivan indices of the generalized hierarchical product of graphs

The vertex Padmakar-Ivan index of a graph G, denoted by PI"v(G), is the sum over all edges uv of G of the number of vertices which are not equidistant from u and v. Similarly, edge Padmakar-Ivan index of a graph G, denoted by PI"e(G), is the sum over all edges uv of G of the number of edges which are not equidistant from u and v. In this paper, PI"v and PI"e of the generalized hierarchical product of two connected graphs are obtained. Using the results obtained here, some known results are deduced as corollaries. Finally, we obtain the PI"v and PI"e of the zig-zag polyhex nanotube TUHC"6[2n,2] and hexagonal chain L"n.