Summary: The Randić index is one of the most important molecular topological indices, and becomes a popular topic of research in mathematics and mathematical chemistry. The sharp upper and lower bounds of Randić index of trees, unicyclic graphs and bicyclic graphs have been obtained. Furthermore, the minimal graphs of trees, unicyclic graphs and bicyclic graphs on Randić index have been characterized. In addition, the lower bounds of Randić index of cacti and corresponding extremal graphs have been described. In this paper, we analyze the degrees of vertices of the edges in cacti, define the symmetric edges and the asymmetric edges, and characterize some transformations. Based on these definitions, according to the discussion of the maximum degree of vertices in the cacti graphs, the asymmetric edge structures of the extremal graphs with the top five Randić indexes in the \(n\)-th order cacti graphs with a given number of circles are obtained.