The extremal graphs of order trees and their topological indices

Recently, D. Vuki$\check{c}$evi$\acute{c}$ and J. Sedlar in \cite{Vuki} introduced an order "$\preceq$" on $\mathcal{T}_n$, the set of trees on $n$ vertices, such that the topological index $F$ of a graph is a function defined on the order set $\langle\mathcal{T}_n,\preceq\rangle$. It provides a new approach to determine the extremal graphs with respect to topological index $F$. By using the method they determined the common maximum and/or minimum graphs of $\mathcal{T}_n$ with respect to topological indices of Wiener type and anti-Wiener type. Motivated by their researches we further study the order set $\langle\mathcal{T}_n,\preceq\rangle$ and give a criterion to determine its order, which enable us to get the common extremal graphs in four prescribed subclasses of $\langle\mathcal{T}_n,\preceq\rangle$. All these extremal graphs are confirmed to be the common maximum and/or minimum graphs with respect to the topological indices of Wiener type and anti-Wiener type. Additionally, we calculate the exact values of Wiener index for the extremal graphs in the order sets $\langle\mathcal{C}(n,k),\preceq\rangle$, $\langle\mathcal{T}_{n}(q),\preceq\rangle$ and $\langle\mathcal{T}_{n}^\Delta,\preceq\rangle$.