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This paper is concerned with the general atom-bond sum-connectivity index A B S γ , which is a generalization of the recently proposed atom-bond sum-connectivity index, where γ is any real number. For a connected graph G with more than two vertices, the number A B S γ (G) is defined as the sum of (1 − 2 (d x + d y) − 1) γ over all edges x y of the graph G, where d x and d y represent the degrees of the vertices x and y of G, respectively. For − 10 ≤ γ ≤ 10 , the significance of A B S γ is examined on the data set of twenty-five benzenoid hydrocarbons for predicting their enthalpy of formation. It is found that the predictive ability of the index A B S γ for the selected property of the considered hydrocarbons is comparable to other existing general indices of this type. The effect of the addition of an edge between two non-adjacent vertices of a graph under A B S γ is also investigated. Furthermore, several extremal results regarding trees, general graphs, and triangle-free graphs of a given number of vertices are proved. [ABSTRACT FROM AUTHOR]