Abstract. The atom-bond connectivity index of a graph G(ABC indexfor short) is defined as the summation of quantitiesq d(u)+d(v)−2d(u)d(v) overall edges of G. A cactus graph is a connected graph in which every blockis an edge or a cycle. The aim of this paper is to obtain the first andsecond maximum values of the ABC index among all n vertex cactusgraphs. 1. IntroductionSuppose G is a simple connected graph with vertex and edge sets V (G) andE(G), respectively. A block of G is a maximal connected subgraphof G withoutcut-vertex. A cactus is a connected graph in which every block is an edge or acycle [18, p. 160]. These are connected graphs in which each edge belongs toat most one cycle. An example of a cactus graph is depicted in Figure 1.Figure 1. Examples of cactus graphs.Cactus graphs have several applications in computer science and biologyand so it is a topic of interest among many researchers in different scientificdisciplines. In [1, 6], it is proved that some graph problems which are NP-hardfor general graphs can be solved in polynomial time for cacti. On the otherhand, in [15] a number of combinatorial optimization problems are presented