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Given a connected graph $G$, Kemeny's constant $\mathcal{K}({G})$ measuresthe average travel time for a random walk to reach a randomly selected vertex.It is known that when an edge is added to $G$, the value of Kemeny's constantmay either decrease, increase, or stay the same. In this paper, we present aquantitative analysis of this behaviour when the initial graph is a tree with$n$ vertices. We prove that when an edge is added into a tree on $n$ vertices,the maximum possible increase in Kemeny's constant is roughly $\frac{2}{3}n,$while the maximum possible decrease is roughly $\frac{3}{16}n^2$. We alsoidentify the trees, and the edges to be added, that correspond to the maximumincrease and maximum decrease. Throughout, both matrix theoretic and graphtheoretic techniques are employed.

Edge Addition and the Change in Kemeny's Constant