The extremal graphs of order trees and their topological indices

Recently, D. Vukičević and J. Sedlar in [1] introduced an order “⪯” on Tn, the set of trees on n vertices, such that the topological index F of a graph is a function defined on the order set 〈Tn,⪯〉. 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 Tn with respect to topological indices of Wiener type and anti-Wiener type. Motivated by their researches we further study the order set 〈Tn,⪯〉 and give a criterion to determine its order, which enable us to get the common extremal graphs in four prescribed subclasses of 〈Tn,⪯〉. 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 〈C(n,k),⪯〉,〈Tn(q),⪯〉 and 〈TnΔ,⪯〉.