The nine smallest hyper-Wiener indices of trees and the eight smallest hyper-Wiener (wiener) indices of unicyclic graphs

Abstract

The seven smallest hyper-Wiener indices of trees on n ≥ 17 vertices have been determined. In this paper, the eighth to ninth smallest hyper-Wiener indices of trees on n ≥ 18 vertices are identified. Moreover, the eight smallest Wiener indices, and the eight smallest hyper-Wiener indices of unicyclic graphs on n ≥ 13 vertices are also determined, and we find that the unicyclic graphs which reach the i-th smallest Wiener indices even share the i-th smallest hyper-Wiener indices of unicyclic graphs on n ≥ 13 vertices for 1 ≤ i ≤ 8.