The Braess' Paradox for Pendant Twins

Abstract

The Kemeny's constant \(\kappa(G)\) of a connected undirected graph \(G\) can be interpreted as the expected transit time between two randomly chosen vertices for the Markov chain associated with \(G\). In certain cases, inserting a new edge into \(G\) has the counter-intuitive effect of increasing the value of \(\kappa(G)\). In the current work we identify a large class of graphs exhibiting this "paradoxical" behavior - namely, those graphs having a pair of twin pendant vertices. We also investigate the occurrence of this phenomenon in random graphs, showing that almost all connected planar graphs are paradoxical. To establish these results, we make use of a connection between the Kemeny's constant and the resistance distance of graphs.