Extremal graphs with respect to generalized ABC index

The generalized ABC index of a graph G, denoted by ABC(alpha)(G), is defined as the sum of weights (d(i)+d(j)-2/d(i)d(j))(alpha) over all edges v(i)v(j) of G, where a is an arbitrary non-zero real number, and d(i) is the degree of vertex v(i) 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 alpha < 0 or 0 < alpha <= 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 ABCa value for alpha < 0 among all connected graphs with given order and vertex connectivity, edge connectivity, or matching number. Our work extends some previously known results. (C) 2018 Elsevier B.V. All rights reserved.