Random Walks on Rooted Trees

For arbitrary positive integers h and m, we consider the family of all rooted trees of height h having exactly m vertices at distance h from the root. We refer to such trees as (h,m)-trees. For a tree T from this family, we consider a simple random walk on T which starts at the root and terminates when it visits one of the m vertices at distance h from the root. Consider the problem of finding a tree in the family on which the expected time of a random walk is minimal (an extremal tree). In this paper we present some properties of extremal trees for arbitrary h and m, completely describe extremal (2,m)and (3,m)-trees, describe a lower bound for the expected time of any (4,m)-tree, and refute the conjecture that the complete binary tree is extremal in the class of all (h, 2h)-trees with the degree of the root at least 2. Introduction All missing definitions related to probability and Markov chains can be found in [2], and those related to graph theory can be found in [8]. Let the greatest distance of any vertex from the root, the height of T , be h. A vertex at a distance k, 0 ≤ k ≤ h, from the root is said to be at tier k. An (h,m)-tree is a rooted tree of height h having exactly m vertices at tier h. Let Ω(T ) be a set of all walks W = v0v1 . . . vn in T such that v0 is the root, vn is at tier h, and vi is at tier h implies i = n. Let dT (u) denote the degree of a vertex u in T , i.e., the number of edges of T incident to u. We define the probability Pr{W} of a walk W ∈ Ω(T ) to be equal to the product of [dT (v)]−1 for all vertices of a walk W , excluding the last. Let X be a random variable representing the length of a walk in Ω(T ), and let E[X] be the expected value of X, i.e., E[X] = ∑ W∈Ω(T ) `(W ) Pr{W}, and we will refer to it as the expected time of a random walk. In this paper, we are concerned with the following: Problem Given two positive integers h and m, we wish to find an (h,m)-tree on which the expected time of a random walk is minimal. We refer to such a tree as being extremal. The above was considered by Lee [3] for the class of “spherically symmetric trees,” that is, trees in which the degrees of all vertices at the same tier are the same. This reduces the analysis of the problem to that of a Markov process on a path of length h. Let us describe the case in which a complete solution (for spherically symmetric (h,m)-trees) was obtained. Suppose there exists a ∗Partially supported by the Undergraduate Research Program of the University of Delaware.