On the general atom-bond sum-connectivity index

Abstract

<abstract><p>This paper is concerned with a generalization of the atom-bond sum-connectivity (ABS) index, devised recently in [A. Ali, B. Furtula, I. Redžepović, I. Gutman, Atom-bond sum-connectivity index, <italic>J. Math. Chem.</italic>, <bold>60</bold> (2022), 2081-2093]. For a connected graph $ G $ with an order greater than $ 2 $, the general atom-bond sum-connectivity index is represented as $ ABS_\gamma(G) $ and is defined as the sum of the quantities $ (1-2(d_x+d_y)^{-1})^{\gamma} $ over all edges $ xy $ of the graph $ G $, where $ d_x $ and $ d_y $ represent the degrees of the vertices $ x $ and $ y $ of $ G $, respectively, and $ \gamma $ is any real number. For $ -10\le \gamma \le 10 $, the significance of $ ABS_\gamma $ is examined on the data set of octane isomers for predicting six selected physicochemical properties of the mentioned compounds; promising results are obtained when the approximated value of $ \gamma $ belongs to the set $ \{-3, 1, 3\} $. The effect of the addition of an edge between two non-adjacent vertices of a graph under $ ABS_\gamma $ is also investigated. Moreover, the graphs possessing the maximum value of $ ABS_{\gamma} $, with $ \gamma > 0 $, are characterized from the set of all connected graphs of a fixed order and a fixed (ⅰ) vertex connectivity not greater than a given number or (ⅱ) matching number.</p></abstract>