The access time of random walks on trees with given partition

Abstract

Denote by T(s,t) the set of trees, whose vertex set can be partitioned into two independent sets of sizes s and t respectively. Given a tree T with stationary distribution π and a vertex v∈T, the access time HT(π,v) is the expected length of optimal stopping rules from π to v. In this paper, we get a sharp upper bound for maxv∈THT(π,v) and a sharp lower bound for minv∈THT(π,v) among T(s,t), respectively. The corresponding extremal graphs are also obtained. As a byproduct, it is proved that the path Pn maximizes maxv∈THT(π,v) and the star K1,n−1 minimizes minv∈THT(π,v) among all trees on n vertices.