Padmakar-Ivan index of power graphs with applications in silicon structures

In this study, we focus on the Padmakar-Ivan (PI) index, a molecular descriptor that quantifies the structural characteristics of chemical networks on the basis of their vertex distances. The main objective of this work is to develop efficient approaches for computing the PI index, particularly for graph powers and complex network structures. We begin by formulating an algorithm for computing the PI index of general graphs and extend it to the $k^{\text{th}}$ power of a graph using a distance-based framework. Furthermore, we introduce a novel clique cut method that establishes a theoretical foundation for analyzing and computing the PI index in intricate silicate and silicon-based frameworks. The proposed techniques significantly simplify and generalize existing computational procedures.