Randić指数是一类重要的分子拓扑指数，在数学及化学研究中有重要作用。树图、单圈图以及双圈图的Randić指数的上下界及其极图已有相关的结论。仙人掌图Randić指数的下界及极图已被刻画，而极大图的研究较为复杂。通过对仙人掌图中边的顶点度的分析，定义了对称边和非对称边，并且刻画了图的一些变换。在此基础上，根据仙人掌图中顶点的最大度分情况讨论，得到了给定圈数r的n阶仙人掌图中具有前五大Randić指数的极图中的非对称边结构。
The Randić index was one of the most important molecular topological indices, and became 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 cacti on Randić index and corresponding extremal graphs have been described. In this paper, we analyzed the degrees of vertices of the edges in cacti, defined the symmetric edges and the asymmetric edges, and characterized some transformations. Based on these definitions, we discussed in terms of maximum degree of vertices. In the end, the extremal graphs have been characterized by the asymmetric edges in cacti of n-vertex given the number of circles with the first, the second, the third, the fourth and the fifth maximum Randić index.
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