Extremal graphs with respect to generalized ABC index

The generalized ABC index of a graph G, denoted by ABCα(G), is defined as the sum of weights (di+dj−2didj)α over all edges vivj of G, where α is an arbitrary non-zero real number, and di is the degree of vertex vi of G. In this paper, we first prove that the generalized ABC index of a connected graph will increase with addition of edge(s) if α<0 or 0<α≤1∕2, which provides a useful tool for the study of extremal properties of the generalized ABC index. By means of this result, we then characterize the graphs having the maximal ABCα value for α<0 among all connected graphs with given order and vertex connectivity, edge connectivity, or matching number. Our work extends some previously known results.