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

The vertex Padmakar–Ivan index of a graph GG, denoted by PIv(G), is the sum over all edges uvuv of GG of the number of vertices which are not equidistant from uu and vv. Similarly, edge Padmakar–Ivan index of a graph GG, denoted by PIe(G), is the sum over all edges uvuv of GG of the number of edges which are not equidistant from uu and vv. In this paper, PIv and PIe 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 PIv and PIe of the zig-zag polyhex nanotube TUHC6[2n,2] and hexagonal chain LnLn.