General Sum-Connectivity Index and General Sombor Index of Chemical Trees with Given Order and Branching Vertices

Abstract

This thesis investigates the properties of graph indices with a focus on the extreme values of chemical trees with given order and branching vertices for the maximum general sum-connectivity index and maximum general Sombor index. For a graph G the general sum-connectivity and general Sombor indices are respectively given by χσ(G) = Σ (dG(p) + dG(q))σ and pq∈E(G) SOσ(G) = Σ (d2 G(p) + d2 G(q))σ. pq∈E(G) Here dG(p) represents the degree of a vertex p in a graph G and E(G) is the edge set of G. Graph indices are numerical quantities that capture specific structural at tributes of graphs and have significant implications in various fields like chem istry, biology, and network analysis. The general sum-connectivity and gen eral Sombor indices derived from the degrees of the vertices provide insights into the connectivity of graphs. Research on the general sum-connectivity index has been going on since 2010. However, the general Sombor index is a newly developed topological index introduced by Hu, X. and Zhong, L. in 2022, which gained extensive attention due to its vast applicability. In this thesis, we will determine the extremal values of general sum-connectivity and general Sombor indices of chemical trees with the given order n and branch ing vertices. By examining these indices, we aim to uncover patterns that enhance our understanding of graph structures and their applications.