Maximal first Zagreb connection index of trees with given total domination number

Abstract

Chemical graph theory plays a crucial role in mathematical chemistry by using graph invariants to represent chemical phenomena in a mathematical framework. Topological descriptors, which are graph invariants formed from molecular graph representations of chemical compounds, are employed in QSPR and QSAR studies. The Zagreb connection indices are topological descriptors which are used to analyze graphs based on connection cardinality. They were introduced in 1972 to calculate the total electron energy of alternate hydrocarbons. Recently, they have gained renewed attention due to their ability to provide a more precise correlation for the physicochemical characteristics of different compounds compared to basic Zagreb indices. In this article, we aim to analyze and establish the correlations of the first Zagreb connection index and the total domination number of trees. Our focus is on providing a detailed description of trees having the largest first Zagreb connection index among trees that have a specific total domination number.