The Average Wiener Index of Trees and Chemical Trees

Meir and Moon ( J. Combin. Theory 1970, 8, 99−103) reported a combinatorial formula for the average value of the distance between a pair of vertices in the class of all labeled trees with a fixed number (= n ) of vertices. From this result an expression for the average Wiener index 〈 W n 〉 lab of labeled n -vertex trees follows immediately. We show that both the average Wiener index 〈 W n 〉 of nonlabeled n -vertex trees and the average Wiener index 〈 W n 〉 ch of nonlabeled n -vertex chemical trees having n ≤ 20 vertices are proportional to 〈 W n 〉 1ab, with proportionality constants around 0.927 and 0.990, respectively. Analogous results are obtained for the Hosoya polynomial.