The asymptotic value of Randic index for trees

Details The asymptotic value of Randic index for trees The asymptotic value of Randic index for trees

2010-03-25

arxiv.org/abs/1003.4810v1

Mathematics

Combinatorics

12 pages

Scientific paper

Let $\mathcal{T}_n$ denote the set of all unrooted and unlabeled trees with $n$ vertices, and $(i,j)$ a double-star. By assuming that every tree of $\mathcal{T}_n$ is equally likely, we show that the limiting distribution of the number of occurrences of the double-star $(i,j)$ in $\mathcal{T}_n$ is normal. Based on this result, we obtain the asymptotic value of Randi\'c index for trees. Fajtlowicz conjectured that for any connected graph the Randi\'c index is at least the 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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