This paper is concerned with the general atom-bond sum-connectivity index ABSγ, 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 ABSγ(G) is defined as the sum of (1−2(dx+dy)−1)γ over all edges xy of the graph G, where dx and dy represent the degrees of the vertices x and y of G, respectively. For −10≤γ≤10, the significance of ABSγ 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 ABSγ 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 ABSγ is also investigated. Furthermore, several extremal results regarding trees, general graphs, and triangle-free graphs of a given number of vertices are proved.