on generalized atombond connectivity index of cacti

The generalized atombond connectivity index of a graph g is denoted by abca(g) and defined as the sum of weights ((d(u)+d(v)2)/d(u)d(v))aa$ over all edges uv∊g. a cactus is a graph in which any two cycles have at most one common vertex. in this paper, we compute sharp bounds for  abca 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.