The Braess' Paradox for Pendant Twins

TL;DR

This paper explores the counter-intuitive phenomenon where adding edges to certain graphs, especially those with twin pendant vertices, increases the expected transit time, revealing paradoxical behaviors in specific graph classes.

Contribution

It identifies a broad class of graphs with twin pendant vertices that exhibit the paradoxical increase in Kemeny's constant upon adding edges and analyzes this in random planar graphs.

Findings

Graphs with twin pendant vertices show increased transit times when edges are added.

Almost all connected planar graphs exhibit this paradoxical behavior.

A connection between Kemeny's constant and resistance distance is utilized.

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.