Padmakar Ivan index for some classes of cycle graphs and perfect graphs

Abstract

The topological index of a graph is a numerical parameter that gives some characterization of its structure. The topological index is a structural invariant. It is a type of molecular descriptor for a chemical compound. Topological index is one of the key areas of research in graph theory and mathematical chemistry. There are different types of topological indices. Many of them have found applications to model chemical, pharmaceutical, and other properties of molecules. The Wiener index, proposed by Harry Wiener in 1947, was the first topological index used to model the chemical properties of molecules. Padmakar Khadikar introduced the Padmakar-Ivan (PI) index of graphs. Later, Khalifeh introduced a vertex version of the PI index. In order to increase the diversity of bipartite graphs, Ilic introduced the weighted PI ´ index. This thesis deals mainly with the PI index (vertex PI) and the weighted PI index. The dissertation starts by studying the nature of these indices for specific classes of graphs, such as powers of paths, cycles, and their complements. We establish a relationship between the PI and weighted PI indices for regular graphs. It allows for calculating the weighted PI index for the power of a cycle. We obtain the PI index of unicyclic and bicyclic graphs. Two different methods for calculating the PI index of a graph are introduced, and these methods are used to calculate the PI index for a variety of perfect graphs. These two methods are computationally easier than the traditional method for some classes of graphs. Using these methods, we obtained the PI index for several classes of perfect graphs, such as co-bipartite graphs, line graphs of certain classes of graphs, chordal graphs, block graphs, and split graphs. Also, we obtained the PI index for some classes of prismatic graphs and cactus graphs. The thesis concludes by using our results to find the formula for the PI index of some chemical networks, including octahedral networks, tetrahedral networks, and oxide networks. We have demonstrated..