Kemeny's constant and Braess cliques in graphs

Abstract

Kemeny's constant is used as a measure of the average travel time on a graph. Braess' paradox for graphs is the observation that in some graphs, when an edge is added, Kemeny's constant increases. We introduce the notion of a Braess clique $K_\ell$, a clique that when inserted into a graph on an independent set of $\ell$ vertices, will create an increase in Kemeny's constant. In this context, a Braess edge is a Braess $K_2$. We provide examples of graphs that have a Braess $K_\ell$ for $\ell\geq 3$. We observe that almost every connected planar labelled graph has a Braess $K_\ell$ for each $\ell\geq 3$. We also explore the relationship between Braess edges and Braess cliques in graphs.