A 1-Separation Formula for the Graph Kemeny Constant and Braess Edges

TL;DR

This paper introduces a simple method to compute Kemeny's constant for 1-separable graphs using electrical resistance, identifies extremal graphs, and characterizes Braess sets that increase the Kemeny constant.

Contribution

It provides a new 1-separation formula for Kemeny's constant, simplifies calculations for specific graph classes, and generalizes the concept of Braess edges to Braess sets.

Findings

Path graph maximizes Kemeny's constant among trees.

Explicit formulas for barbell graphs' Kemeny's constant.

Conditions for the existence of Braess sets in certain graphs.

Abstract

Kemeny's constant of a simple connected graph $G$ is the expected length of a random walk from $i$ to any given vertex $j \neq i$. We provide a simple method for computing Kemeny's constant for 1-separable via effective resistance methods from electrical network theory. Using this formula, we furnish a simple proof that the path graph on $n$ vertices maximizes Kemeny's constant for the class of undirected trees on $n$ vertices. Applying this method again, we simplify existing expressions for the Kemeny's constant of barbell graphs and demonstrate which barbell maximizes Kemeny's constant. This 1-separation identity further allows us to create sufficient conditions for the existence of Braess edges in 1-separable graphs. We generalize the notion of the Braess edge to Braess sets, collections of non-edges in a graph such that their addition to the base graph increases the Kemeny constant. We characterize Braess sets in graphs with any number of twin pendant vertices, generalizing work of Kirkland et.~al.~\cite{kirkland2016kemeny} and Ciardo \cite{ciardo2020braess}.