A graph G consists of a set of vertices connected in pairs by edges. Two vertices connected by an edge are called neighbors. A random walk on G is a sequence of vertices, where the next vertex is chosen randomly from the neighbors of the current vertex. Under mild assumptions, the distribution of a walk's current location converges to the stationary distribution. In this distribution, the probability of being at a given vertex is proportional to the size of its neighbor set. The expected time to convergence is called a mixing measure of G. There are a variety of mixing measures, depending on which vertex is chosen for the starting state of the walk. For walks starting at the worst initial vertex, an average initial vertex, and the best initial vertex, the mixing measures are called the "mixing time," the "reset time" and the "best mixing time," respectively.
We study random walks on trees on n vertices. For a given mixing measure, we aim to determine what tree structure maximizes this value, and what tree structure minimizes this value. Previous research by Beveridge and Wang (Macalester 2009) resolved this question for the mixing time and the reset time. For each of these measures, the n-star uniquely achieves the minimum, and the n-path uniquely achieves the maximum. Herein, we resolve this question for the best mixing time. Once again, the n-star is the minimizing structure. However, we prove an unexpected result for the maximizing structure. While the maximizing structure for even n is indeed the n-path, the maximizing-structure for odd n is in fact an (n — 1)-path with a single leaf attached to one of the central vertices.
Youngblood, Jeanmarie, "The Best Mixing Time for Trees" (2012). Mathematics, Statistics, and Computer Science Honors Projects. 27.
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