Computing Kemeny's constant for a barbell graph

Abstract

In a graph theory setting, Kemeny’s constant is a graph parameter which measures a weighted average of the mean first passage times in a random walk on the vertices of the graph. In one sense, Kemeny’s constant is a measure of how well the graph is ‘connected’. An explicit computation for this parameter is given for graphs of order n consisting of two large cliques joined by an arbitrary number of parallel paths of equal length, as well as for two cliques joined by two paths of different length. In each case, Kemeny’s constant is shown to be O(n3), which is the largest possible order of Kemeny’s constant for a graph on n vertices. The approach used is based on interesting techniques in spectral graph theory and includes a generalization of using twin subgraphs to find the spectrum of a graph.

Kemeny-based testing for COVID-19

“…A characteristic polynomial interpretation of K is also given in [ 20 , 21 ]. To the best of our knowledge the link to the characteristic polynomial and the Markov transition matrix, more precisely , was first given in [ 20 ].…”

“…To the best of our knowledge the link to the characteristic polynomial and the Markov transition matrix, more precisely , was first given in [ 20 ]. A further derivation is given in [ 21 ], this time using the associated adjacency matrix. However, the derivation given here expresses K from the derivative of the characteristic polynomial associated with .…”

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

“…Furthermore, we demonstrate several uses of this 1-separation formula. First, we give simple expressions for Kemeny's constant of barbell graphs, which were studied in [5]. Barbell graphs are of interest in the study of Kemeny's constant, as they are believed, based on empirical computation, to maximize Kemeny's constant among graphs on a given number of vertices.…”

“…Barbell graphs are of interest in the study of Kemeny's constant, as they are believed, based on empirical computation, to maximize Kemeny's constant among graphs on a given number of vertices. From [5], it is known that certain barbells on n vertices have Kemeny's constant on the order of n 3 , and that order n 3 is the largest Kemeny's constant can be. Our 1separation formula allows us to give an exact expression for Kemeny's constant in barbell graphs that is much simpler than that found in [5], and we are able to determine what barbell has the largest Kemeny's constant among all barbell graphs on n vertices.…”

“…From [5], it is known that certain barbells on n vertices have Kemeny's constant on the order of n 3 , and that order n 3 is the largest Kemeny's constant can be. Our 1separation formula allows us to give an exact expression for Kemeny's constant in barbell graphs that is much simpler than that found in [5], and we are able to determine what barbell has the largest Kemeny's constant among all barbell graphs on n vertices. Second, we use our formula to prove that, among all trees on n vertices, the path graph has the largest Kemeny's constant.…”