Extremal graphs with respect to generalized [formula omitted] index.

The generalized A B C index of a graph G , denoted by A B C α ( G ) , is defined as the sum of weights ( d i + d j − 2 d i d j ) α over all edges v i v j of G , where α 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 A B C 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 A B C index. By means of this result, we then characterize the graphs having the maximal A B C α 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. [ABSTRACT FROM AUTHOR]