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

TL;DR

This paper characterizes the extremal weighted trees that maximize or minimize average hitting time and Kemeny's constant, using forest formulas, among trees with fixed edge weight multisets.

Contribution

It provides a complete characterization of weighted trees that optimize average hitting time and Kemeny's constant, identifying polarized paths and a weighted star as extremal structures.

Findings

Polarized paths maximize average hitting time.

Weighted star minimizes average hitting time.

Similar extremal trees are identified for Kemeny's constant.

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 obtain a similar characterization of the $\kappa$-maximizing and $\kappa$-minimizing elements among such a collection of weighted trees. Our proofs are based on the forest formulas for $\alpha$ and $\kappa$.