A New Method for Computing the Vertex PI Index with Applications to Special Classes of Graphs

The Padmakar-Ivan (PI) index of a graph G is given by 𝑃𝐼⁡(𝐺)=∑𝑒∈𝐸⁡(𝐺)(|𝑉⁡(𝐺)|−𝑁𝐺(⁢𝑒)), where 𝑁𝐺⁡(𝑒) is the number of equidistant vertices for the edge e. This paper presents a triangle cover for a graph, along with a novel method for finding the PI index using this cover. The technique is used to examine some chemical networks, including octahedral and oxide networks, leading to the determination of the exact formula for their PI indices. Additionally, the approach is used to study perfect graphs such as prismatic and chordal graphs. Finally, the PI index of chordal graphs with a diameter of two is investigated, focusing on their induced subgraphs.