The asymptotic value of the Randić index for trees

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 Randić index for trees. Fajtlowicz conjectured that for any connected graph the Randić 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.

The asymptotic value of the zeroth-order Randić index and sum-connectivity index for trees

“…We refer the readers to [12] and [27] for details. In [10,26], the authors established this property in general for more complicated structures.…”

The asymptotic number of non-isomorphic rooted trees obtained by rooting a tree

“…Clearly, if one shows that b = 0, then the distribution is normal. For some special patterns, such as a star (or a node with a given degree) pattern [5], a double-star pattern [10], and a path pattern [9], the corresponding limiting distributions were proved to be normal. For some previous work we refer to Robinson and Schwenk [13].…”