Monotonicity conjecture on permanents of doubly stochastic matrices

MONOTONICTTY CONJECTURE ON PERMANENTS OF DOUBLY STOCHASTIC MATRICES i

A stronger version of the van der Waerden permanent conjecture asserts that if J„ denotes the n X n matrix all of whose entries are \/n and A is any fixed matrix on the boundary of the set ofnxn doubly stochastic matrices, then per(A/4 + (1 — \)Jn) as a function of X is nondecreasing in the interval [0, 1]. In this paper, we elucidate the relation of this assertion to some other conjectures kno...

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...

On a Conjecture concerning Doubly Stochastic Matrices

Minimum permanents on two faces of the polytope of doubly stochastic matrices∗

We consider the minimum permanents and minimising matrices on the faces of the polytope of doubly stochastic matrices whose nonzero entries coincide with those of, respectively, Um,n = [ In Jn,m Jm,n 0m ] and Vm,n = [ In Jn,m Jm,n Jm,m ] . We conjecture that Vm,n is cohesive but not barycentric for 1 < n < m + √ m and that it is not cohesive for n > m + √ m. We prove that it is cohesive for 1 <...

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...