DETOUR PADMAKAR IVAN INDICES OF HARMONIC GRAPHS

Abstract

Graph theory is a fascinating and intellectually engaging area within the broader scope of Discrete Mathematics, providing abundant opportunities to explore a wide range of proof techniques. The concepts and findings from graph theory have far-reaching implications, making it a versatile tool with applications in various fields, including computer science, physics, biology, and social network analysis. This vast field of study classifies graphs based on particular structural characteristics. Specifically, a graph G is called a Pseudo-regular graph if all its vertices share the same average degree, indicating a consistent level of connectivity across the graph and resulting in a highly symmetric structure. In contrast, if the average degree of vertices varies across the graph, it is categorized as a Harmonic graph. Harmonic graphs feature a more intricate structure, where the degree of connectivity between vertices is not uniform. The differentiation between pseudo-regular and harmonic graphs underscores the diversity found within graph types and highlights the rich theoretical foundation that graph theory provides for the analysis of complex networks and structures. The Detour Padmakar-Ivan (DPI) of a graph G is defined as and edges equidistant from both ends of the edges e = uv are not counted.  Similar to the vertex version of Detour PI index, another important index Detour Co-PI index of G which is defined as .   In this paper computation of the Detour Pi , Detour Co-Pi Index in Harmonic  graphs are proposed.