Abstract
Let T n denote the set of all unrooted and unlabeled trees with n vertices, and ( i , j ) a double-star. By assuming that every tree of T n is equally likely, we show that the limiting distribution of the number of occurrences of the double-star ( i , j ) in T n is normal. Based on this result, we obtain the asymptotic value of the Randić index for trees. Fajtlowicz conjectured that for any connected graph G the Randić index of G is at least its average distance. Using this asymptotic value, we show that this conjecture is true not only for almost all connected graphs but also for almost all trees.
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