Edge addition and the change in kemeny’s constant

Given a connected graph \(G\), Kemeny’s constant \(\mathcal{K}(G)\) measures the 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 constant may either decrease, increase, or stay the same. In this paper, we present a quantitative 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 also identify the trees, and the edges to be added, that correspond to the maximum increase and maximum decrease. Throughout, both matrix theoretic and graph theoretic techniques are employed.

Recommended citation: Kirkland, S., Li, C., McAlister, J.S., and Zhang, X. (2023) Edge Addition and the Change in Kemeny's Constant. Discrete Applied Mathematics. https://doi.org/10.1016/j.dam.2025.04.031
Download Paper | Link to Paper