The asymptotic values of the general Zagreb and Randic indices of trees with bounded maximum degree

arXiv:1004.1778v1 [math.CO] 11 Apr 2010 The asymptotic values of the general Zagreb and Randi´c indices of trees with bounded maximum degree∗ Xueliang Li, Yiyang Li Center for Combinatorics and LPMC-TJKLC Nankai University, Tianjin 300071, China Abstract Let T ∆ n denote the set of trees of order n, in which the degree of each vertex is bounded by some integer ∆. Suppose that every tree in T ∆ n is equally likely. We show that the number of vertices of degree j in T ∆ n is asymptotically normal with mean (µj + o(1))n and variance (σj + o(1))n, where µj, σj are some constants. As a consequence, we give estimate to the value of the general Zagreb index for almost all trees in T ∆ n . Moreover, we obtain that the number of edges of type (i, j) in T ∆ n also has mean (µij + o(1))n and variance (σij + o(1))n, where an edge of type (i, j) means that the edge has one end of degree i and the other of degree j, and µij, σij are some constants. Then, we give estimate to the value of the general Randi´c index for almost all trees in T ∆ n . Keywords: generating function, tree, normal distribution, asymptotic value, general Zagreb index, general Randi´c index. AMS subject classification 2010: 05C05, 05C12, 05C30, 05D40, 05A15, 05A16, 92E10 1 Introduction In this paper, we mainly consider trees, in which the degree of each vertex is bounded by some integer ∆. If ∆= 1, 2, the cases are trivial. Thus, we suppose ∆≥3 throughout this paper. Let T ∆ n denote the set of trees with n vertices. We suppose that every tree in T ∆ n is equally likely and Xn is a random variable, such as the number of vertices of degree j, or the number of edges of type (i, j), each having one end of degree i and the other of degree ∗Supported by NSFC No.10831001, PCSIRT and the “973” program. 1 j. It is easy to see that Xn can take at most |T ∆ n | distinct values. We first introduce two generating functions. Setting tn = |T ∆ n |, we have t(x) = X n≥1 tnxn, t(x, u) = X n≥1,k≥0 tn,kxnuk, where tn,k denotes the number of trees in T ∆ n such that Xn = k. Therefore, the probability of Xn can be defined as Pr[Xn = k] = tn,k tn . Note that t(x, 1) = t(x). In [11], it is showed that tn is asymptotically equal to τ · x−n 0 n5/2, where τ and x0 are constants with x0 ≤1/2. In conjunction with the generating functions and asymptotic analysis, in [4] and [13] the authors investigated the limiting distribution of the number of vertices of given degree j for trees without degree restriction. By the same method, many results have been established for other variables, such as the number of a given path or pattern (see [8]) for rooted trees, planar trees, labeled trees et al. However, all the statements showed that the limiting distributions are normal. We refer the readers to [2] and [8] for further details. In this sequel, we follow the method used in [2] and [4] to obtain that the distribution of the number of vertices of degree j for trees in T ∆ n is also asymptotically normal with mean (µj + o(1))n and variance (σj + o(1))n. Then, we give estimate to the value of the general Zagreb index for almost all trees in T ∆ n . However, for the number of edges of type (i, j), we only get a weak statement which can not show that the limiting distribution is normal. Nevertheless, we still can use it to obtain the asymptotical value of the general Randi´c index for almost all trees in T ∆ n . The definitions of the general Zagreb index and general Randi´c index will be given in next sections. Many results have been obtained for the two parameters. We refer the readers to [9] and [10] for a detailed survey. In this paper we will show that for the random space T ∆ n , each of the indices has a value of Θ(n) for almost all trees. Section 2 is devoted to a systematic treatment of the number of vertices of degree j and the general Zagreb index. In Section 3, we investigate the number of edges of type (i, j) and the general Randi´c index. 2 The number of vertices of degree j In this section, we first consider the the limiting distribution of the number of vertices of degree j in T ∆ n . Then, as an immediate consequence, we get the asymptotic value of the general Zagreb index for almost all trees in T ∆ n . 2 In what follows, we introduce some terminology and notation which will be used in the sequel. For the others not defined here, we refer to book [7]. Analogous to trees, we introduce generating functions for rooted trees and planted trees. Let R∆ n denote the set of rooted trees of order n with degrees bounded by an integer ∆. Setting rn = |R∆ n |, we have r(x) = X n≥1 rnxn and r(x, u) = X n≥1,k≥0 rn,kxnuk, where rn,k denotes the number of trees in R∆ n such that Xn equals k. A planted tree is formed by adding a vertex to the root of a rooted tree. The new vertex is called the plant, and we never count it in the sequel. Analogously, let P∆ n denote the set of planted trees with n vertices of bounded maximum degree ∆. Setting pn = |P∆ n |, we have p(x) = X n≥1 pnxn and p(x, u

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