# A 1-separation formula for the graph Kemeny constant and Braess edges

@article{Faught2021A1F,
title={A 1-separation formula for the graph Kemeny constant and Braess edges},
author={Nolan Faught and Mark Kempton and Adam Knudson},
journal={Journal of Mathematical Chemistry},
year={2021},
pages={1 - 21}
}
• Published 2021
• Mathematics
• Journal of Mathematical Chemistry
Kemeny’s constant of a simple connected graph G is the expected length of a random walk from i to any given vertex j≠i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \ne i$$\end{document}. We provide a simple method for computing Kemeny’s constant for 1-separable graphs via effective resistance methods from… Expand
1 Citations

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#### References

SHOWING 1-10 OF 26 REFERENCES
Resistance distance in straight linear 2-trees
• Computer Science, Mathematics
• Discret. Appl. Math.
• 2019
These results for straightlinear 2-trees along with an example of a bent linear 2-tree and empirical results for additional graph classes convincingly demonstrate that resistance distance should not be discounted as a viable method for link prediction in geometric graphs. Expand
On Kemeny's constant for trees with fixed order and diameter
• Mathematics
• 2020
Kemeny's constant $\kappa(G)$ of a connected graph $G$ is a measure of the expected transit time for the random walk associated with $G$. In the current work, we consider the case when $G$ is a tree,Expand
Computing Kemeny's constant for a barbell graph
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,Expand
Kirchhoff index, multiplicative degree-Kirchhoff index and spanning trees of the linear crossed polyomino chains
• Mathematics
• 2018
Let $G_n$ be a linear crossed polyomino chain with $n$ four-order complete graphs. In this paper, explicit formulas for the Kirchhoff index, the multiplicative degree-Kirchhoff index and the numberExpand
KEMENY'S CONSTANT AND AN ANALOGUE OF BRAESS' PARADOX FOR TREES ∗
• Mathematics
• 2016
Given an irreducible stochastic matrix M, Kemenyâs constant K(M) measures the expected time for the corresponding Markov chain to transition from any given initial state to a randomly chosen finalExpand
The Braess' paradox for pendent twins
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 $G$. InExpand
The Kemeny Constant for Finite Homogeneous Ergodic Markov Chains
• Mathematics, Computer Science
• J. Sci. Comput.
• 2010
A new formula for the Kemeny constant is presented and several perturbation results for the constant are developed, including conditions under which it is a convex function and for chains whose transition matrix has a certain directed graph structure. Expand
Broder and Karlin's formula for hitting times and the Kirchhoff Index
• Mathematics
• 2011
We give an elementary proof of an extension of Broder and Karlin's formula for the hitting times of an arbitrary ergodic Markov chain. Using this formula in the particular case of random walks onExpand
Bounds for the Kirchhoff index of regular graphs via the spectra of their random walks
• Mathematics
• 2010
Using probabilistic tools, we give tight upper and lower bounds for the Kirchhoff index of any d-regular N-vertex graph in terms of d, N, and the spectral gap of the transition probability matrixExpand
Graphs and Matrices
• Mathematics
• 1999
In order to store a graph or digraph in a computer, we need something other than the diagram or the formal definition. This something is the adjacency matrix, a matrix of O’s and l’s. The l’sExpand