Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 8 1 2023 03 01 Unicyclic graphs with maximum Randić indices 161 172 14327 10.22049/cco.2021.27230.1216 EN Roslan Hasni UMT, Malaysia Nor Hafizah Md Husin Universiti Pendidikan Sultan Idris Zhibin Du School of Mathematics and Statistics, Zhaoqing University, Zhaoqing 526061, China Journal Article 2021 04 27 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 $n\ge 5$, the fourth for $n\ge 7$ and the fifth for $n\ge 8$ 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 $n\ge 9$ and the eighth for $n\ge 10$ 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. https://comb-opt.azaruniv.ac.ir/article_14327_6555b5e7ba897c85af34e06ed66c2deb.pdf