A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices

Doubly stochastic matrices of trees

In this paper, we obtain sharp upper and lower bounds for the smallest entries of doubly stochastic matrices of trees and characterize all extreme graphs which attain the bounds. We also present a counterexample to Merris’ conjecture on relations between the smallest entry of the doubly stochastic matrix and the algebraic connectivity of a graph in [R. Merris, Doubly stochastic graph matrices I...

Random doubly stochastic tridiagonal matrices

Let Tn be the compact convex set of tridiagonal doubly stochastic matrices. These arise naturally in probability problems as birth and death chains with a uniform stationary distribution. We study ‘typical’ matrices T ∈ Tn chosen uniformly at random in the set Tn. A simple algorithm is presented to allow direct sampling from the uniform distribution on Tn. Using this algorithm, the elements abo...

A note on doubly stochastic graph matrices

A sharp lower bound for the smallest entries, among those corresponding to edges, of doubly stochastic matrices of trees is obtained, and the trees that attain this bound are characterized. This result is used to provide a negative answer to Merris’ question in [R. Merris, Doubly stochastic graph matrices II, Linear Multilin. Algebra 45 (1998) 275–285]. © 2005 Elsevier Inc. All rights reserved....

On a Conjecture concerning Doubly Stochastic Matrices

On A Conjecture Concerning Doubly Stochastic Matrices

In a 2002 paper, Kirkland showed that if T ∈ Rn×n is an irreducible stochastic matrix with stationary distribution vector πT , then for A = I − T , maxj=1,...,n πj‖A j ‖∞ ≥ n−1 n , where Aj , j = 1, . . . , n, are the (n − 1) × (n − 1) principal submatrices of A obtained by deleting the j–th row and column of A. He also conjectured that equality holds in that lower bound if and only if either T...