The extremal graphs of order trees and their topological indices

Recently, D. Vukicevic and J. Sedlar in [1] introduced an order "<=" on 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 < T-n, <=>. 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 T-n with respect to topological indices of Wiener type and anti-Wiener type. Motivated by their researches we further study the order set < T-n, <=> and give a criterion to determine its order, which enable us to get the common extremal graphs in four prescribed subclasses of < T-n,<=>. All these extremal graphs are confirmed to be the com- mon 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),<=>, < T-n(q),<=>, and < T-n(Delta),<=>. (C) 2021 Elsevier Inc. All rights reserved.