MINIMUM PERMANENTS ON DOUBLY STOCHASTIC MATRICES WITH PRESCRIBED ZEROS

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...

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 <...

Maximum permanents of matrices of zeros and ones

Let U(n, T) be the set of ail matrices of O's and l's of order n with exactly r 0's. We obtain an upper bound for the permanent of a matrix in U(n, t). For 0 <r < 2n and for nz-2n C r < n2-n we determine all matrices in U(n, r) with maximum permanent.

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