Unicyclic graphs with maximum Randić indices

The Randi'c index $R(G)$ of a graph $G$ is the sum of the weights $(d_u d_v)^{-frac{1}{2}}$ of all edges $uv$ in $G$, where $d_u$ denotes the degree of vertex $u$. Du and Zhou [On Randi'c indices of trees, unicyclic graphs, and bicyclic graphs, Int. J. Quantum Chem. 111 (2011), 2760--2770] determined the $n$-vertex unicyclic graphs with the third for $nge 5$, the fourth for $nge 7$ and the fifth for $nge 8$ maximum Randi'c indices. Recently, Li et al. [The Randi{' c} indices of trees, unicyclic graphs and bicyclic graphs, Ars Combin. 127 (2016), 409--419] obtained the $n$-vertex unicyclic graphs with the sixth and the seventh for $nge 9$ and the eighth for $nge 10$ maximum Randi'c indices. In this paper, we characterize the $n$-vertex unicyclic graphs with the ninth, the tenth, the eleventh, the twelfth and the thirteenth maximum Randi'c values.