More Results on Extremum Randi{\' c} Indices of (Molecular) Trees

Abstract

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 trees with the third for $n\ge 7$, the fourth for $n\ge 10$, the fifth and the sixth for $n\ge 11$ maximum Randi{\' c} indices. Recently, Li \textit{et al.} [The Randi{\' c} indices of trees, unicyclic graphs and bicyclic graphs, Ars Comb. 127 (2016), 409--419] obtained the $n$-vertex trees with the seventh, the eighth, the ninth and the tenth for $n\ge 11$ maximum Randi{\' c} indices. In this paper, we correct the ordering for the Randi{\' c} indices of trees obtained by Li \textit{et al.}, and characterize the trees with from the seventh to the sixteenth maximum Randi{\' c} indices. The obtained extremal trees are molecular and thereby the obtained ordering also holds for molecular trees.