Abstract
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 〈Wn〉lab of labeled n-vertex trees follows immediately. We show that both the average Wiener index 〈Wn〉 of nonlabeled n-vertex trees and the average Wiener index 〈Wn〉ch of nonlabeled n-vertex chemical trees having n ≤ 20 vertices are proportional to 〈Wn〉1ab, with proportionality constants around 0.927 and 0.990, respectively. Analogous results are obtained for the Hosoya polynomial.
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