Threshold graphs, Kemeny's constant, and related random walk parameters

Abstract

Kemeny's constant measures how fast a random walker moves around in a graph. Expressions for Kemeny's constant can be quite involved, and for this reason, many lines of research focus on graphs with structure that makes them amenable to more in-depth study (for example, regular graphs, acyclic graphs, and 1-connected graphs). In this article, we study Kemeny's constant for random walks on threshold graphs, which are an interesting family of graphs with properties that make examining Kemeny's constant difficult; that is, they are usually not regular, not acyclic, and not 1-connected. This article is a showcase of various techniques for calculating Kemeny's constant and related random walk parameters for graphs. We establish explicit formulae for $\mathcal{K}(G)$ in terms of the construction code of a threshold graph, and completely determine the ordering of the accessibility indices of vertices in threshold graphs.