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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 amenableto more in-depth study (for example, regular graphs, acyclic graphs, and1-connected graphs). In this article, we study Kemeny's constant for randomwalks on threshold graphs, which are an interesting family of graphs withproperties that make examining Kemeny's constant difficult; that is, they areusually not regular, not acyclic, and not 1-connected. This article is ashowcase of various techniques for calculating Kemeny's constant and relatedrandom walk parameters for graphs. We establish explicit formulae for$\mathcal{K}(G)$ in terms of the construction code of a threshold graph, andcompletely determine the ordering of the accessibility indices of vertices inthreshold graphs.

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