On Generalized Atom−Bond Connectivity Index of Cacti

The generalized atom-bond connectivity index of a graph G is denoted by ABCα(G) and defined as the sum of weights (d(u)✚d(v)–2/d)u)d(v))α over all edges uv ∈ G , where d(u) is the degree of the vertex u in G , and α is an arbitrary non-zero real number . A cactus is a graph in which any two cycles have at most one common vertex . In this paper , we compute sharp bounds for ABC α index for cacti of order n with fixed number of cycles and for cacti of order n with given number of pendant vertices . Furthermore , we identify all the cacti that achieve the bounds .