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) | - NG(e)). The weighted PI index of a graph G is given by, PIw(G) = ∑e=(u,v)∈E(G)(dG(u) + dG(v))(|V(G) | - NG(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 kth power of a cycle.