**1. ** A connected graph is a tree if it contains no cycles. Show that every finite tree with at least two vertices must have at least two leaves. (a leaf is a vertex of degree 1).

**2.** A vertex coloring of a graph is called proper if every pair of adjacent vertices are assigned different colors. Consider the following Markov chain on the set of proper 3 colorings of a finite tree : At each step, a vertex of is selected uniformly at random, and is assigned randomly, uniformly one of the colors that does not appear among the neighbors of . Show that this chain is irreducible.

**3.** Let be a transition matrix on a finite set . Suppose that is a stationary distribution for and . Show that