A 1Separation Formula for the Graph Kemeny Constant and Braess Edges
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 1separable 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 1separation identity further allows us to create sufficient conditions for the existence of Braess edges in 1separable graphs. We generalize the notion of the Braess edge to Braess sets, collections of nonedges 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}.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.01061
 Bibcode:
 2021arXiv210801061F
 Keywords:

 Mathematics  Combinatorics;
 05C50