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On the Atom-Bond Connectivity index of cacti

Dong Hawei (Minjiang University, Department of Mathematics, Fuzhou Fujian, China)
Wu Xiaoxia (Minnan Normal University, School of Mathematics and Statistics, Zhangzhou Fujian, China)

The Atom-Bond Connectivity (ABC) index of a connected graph G is defined as ABC(G) = Σuv(E(G)√d(u)+d(v)-2/d(u)d(v), where d(u) is the degree of vertex u in G. A connected graph G is called a cactus if any two of its cycles have at most one common vertex. Denote by G0(n, r) the set of cacti with n vertices and r cycles and G1(n,p) the set of cacti with n vertices and p pendent vertices. In this paper, we give sharp bounds of the ABC index of cacti among G0(n,r) and G1(n,p) respectively, and characterize the corresponding extremal cacti.

Keywords: Atom-Bond Connectivity index, cactus, upper bound