On Weighted PI Index of Graphs

Abstract The vertex PI index of a graph G , denoted by P I 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, the Weighted PI index of a graph G , denoted by P I w ( G ) = ∑ u v = e ∈ E ( G ) ( d G ( u ) + d G ( v ) ) ( n u G ( e ) + n v G ( e ) ) , where d G ( u ) is the degree of the vertex u in G . In this paper, the exact formula for the weighted PI indices of generalized hierarchical product and join of two graphs are obtained. Using the result obtained here, some known result is deduced as corollary. Also, we obtain the weighted PI indices of the zig-zag polyhex nanotube T U H C 6 [ 2 n , 2 ] , hexagonal chain L n , the molecular graph of Truncated Cube, fan and wheel graphs.