We study random walks on trees, where we iteratively move from one vertex to a randomly chosen adjacent vertex. We study two quantities arising in random walks: the hitting time and the mixing time. The hitting time is the expected number of steps to walk between a chosen pair of vertices. The mixing time is the expected number of steps before the distribution of the current state is proportional to its degree. For a fixed tree size, we prove that the star is the unique minimizing structure and the path is the unique maximizing structure for both quantities.
Wang, Meng, "Extremal Random Walks on Trees" (2009). Mathematics, Statistics, and Computer Science Honors Projects. 12.
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