PI and weighted PI indices for powers of paths,cycles,and their complements

Abstract

 Topological indices play a significant role in molecular chemistry, spectral graph theory, network theory, etc. We aim to contribute some new results on PI and weighted PI indices. The vertex PI index of a graph G is given by, PI(G) = ∑_e∈E(G)(|V(G) | - N _G (e)). The weighted PI index of a graph G is given by, PI _w (G) = ∑_{e=(u,v)∈E(G)}(d _G (u) + d _G (v))(|V(G) | - N _G (e)). We obtained the PI and weighted PI indices for powers of paths, cycles, and their complements in this study. Also, for a regular graph, a relationship between PI and weighted PI indices is established, and using this relation the weighted PI index is calculated for k ^th power of a cycle.