On the general atom-bond sum-connectivity index

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, J. Math. Chem., 60 (2022), 2081-2093]. For a connected graph $G$ with an order greater than $2$, the general atom-bond sum-connectivity index is represented as $AB{S}_{\gamma }(G)$ 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 $AB{S}_{\gamma }$ 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 $AB{S}_{\gamma }$ is also investigated. Moreover, the graphs possessing the maximum value of $AB{S}_{\gamma }$, 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.