On average hitting time and Kemeny's constant for weighted trees

Abstract: For a connected graph $G$, the average hitting time $\alpha(G)$ and the Kemeny's constant $\kappa(G)$ are two similar quantities, both measuring the time for the random walk on $G$ to travel between two randomly chosen vertices. We prove that, among all weighted trees whose edge-weights form a fixed multiset, $\alpha$ is maximized by a special type of "polarized" paths and is minimized by a unique weighted star graph. We also give a short proof of the fact that, among all simple trees of a fixed size, $\kappa$ is maximized by the path and is minimized by the star graph. Our proofs are based on the forest formulas for the average hitting time and the Kemeny's constant.