More on Lanzhou and Ad-hoc Lanzhou indices of Graphs

Topological indices are graph-theoretical descriptors that offer a computationally efficient means of predicting the physicochemical properties of chemical compounds, thereby reducing the need for extensive, resource-intensive laboratory experimentation. The Lanzhou index of a simple connected graph [Formula: see text] is defined as [Formula: see text] where [Formula: see text] and [Formula: see text] denote the degree of vertex [Formula: see text] in [Formula: see text] and in its complement graph [Formula: see text], respectively. In this work, we introduce and study the Ad-hoc Lanzhou index, [Formula: see text] obtained by interchanging the roles of the two degree terms in the original definition. We investigate how both the Lanzhou index and the Ad-hoc Lanzhou index respond to structural perturbations of the graph, specifically the addition and removal of edges and vertices, establishing analytical results that characterize these changes. To assess the practical predictive value of these indices, we compare the Lanzhou and Ad-hoc Lanzhou indices against several physicochemical characteristics of twenty-two benzenoid hydrocarbons – a class of compounds of established environmental and toxicological relevance. Our analysis reveals that the Ad-hoc Lanzhou index exhibits a markedly stronger correlation than classical topological indices, achieving a correlation coefficient of [Formula: see text] with the retention index of benzenoid hydrocarbons. These findings suggest that the Ad-hoc Lanzhou index can serve as a reliable, low-cost predictive tool for chemical property estimation, supporting more efficient and sustainable approaches to chemical characterization and structure–property research.