Given an n-vertex graph G, the matrix Ω(G) = (In + L(G))−1 = (ωij) is called the doubly stochastic graph matrix of G, where In is the n × n identity matrix and L(G) is the Laplacian matrix of G. Let ω(G) be the smallest element of Ω(G). Zhang and Wu [X.D. Zhang and J.X. Wu. Doubly stochastic matrices of trees. Appl. Math. Lett., 18:339–343, 2005.] determined the tree T with the minimum ω(T ) am...

The polytope Q, of the convex combinations of the permutation matrices of order n is well known (Birkhoff’s theorem) to be the polytope of doubly stochastic matrices of order n. In particular it is easy to decide whether a matrix of order n belongs to Q,. . check to see that the entries are nonnegative and that all row and columns sums equal 1. Now the permutations z of { 1, 2, . . . . n} are i...

A fundamental step in many data-analysis techniques is the construction of an affinity matrix describing similarities between data points. When points reside Euclidean space, a widespread approach to form by Gaussian kernel with pairwise distances, and follow certain normalization (e.g., row-stochastic or its symmetric variant). We demonstrate that doubly stochastic zero main diagonal (i.e., no...

Given an n-vertex graph G, the matrix Ω(G) = (In + L(G))−1 = (ωij) is called the doubly stochastic graph matrix of G, where In is the n × n identity matrix and L(G) is the Laplacian matrix of G. Let ω(G) be the smallest element of Ω(G). Zhang and Wu [X.D. Zhang and J.X. Wu. Doubly stochastic matrices of trees. Appl. Math. Lett., 18:339–343, 2005.] determined the tree T with the minimum ω(T ) am...

We generalize the classical notion of majorization in Rn to a majorization order for functions defined on a partially ordered set P . In this generalization we use inequalities for partial sums associated with ideals in P . Basic properties are established, including connections to classical majorization. Moreover, we investigate transfers (given by doubly stochastic matrices), complexity issue...

In this study, we give a characterization of all torsion units which are in the unit group of ZS3 integral group ring of symmetric group S3, and classify conjugate classes of these units. We used the group of all doubly stochastic matrices in GL(3,Z) in this classification. The investigation of torsion units is not restricted with this study, and the classification of torsion units of bigger or...

A conditional spatial autoregression (CAR) specifies dependence via a weight matrix. Employing a doubly stochastic weight matrix allows users to interpret the CAR prediction rule as a semiparametric prediction rule and as BLUP with smoothing in addition to other benefits. We examine standard and doubly stochastic weight matrices in the context of an illustrative data set to demonstrate feasibil...

Some special subsets of the set of uniformly tapered doubly stochastic matrices are considered. It is proved that each such subset is a convex polytope and its extreme points are determined. A minimality result for the whole set of uniformly tapered doubly stochastic matrices is also given. It is well known that if x and y are nonnegative vectors of R and x is weakly majorized by y, there exist...

In Ehrhart theory, the h∗-vector of a rational polytope often provides insights into properties of the polytope that may be otherwise obscured. As an example, the Birkhoff polytope, also known as the polytope of real doubly-stochastic matrices, has a unimodal h∗-vector, but when even small modifications are made to the polytope, the same property can be very difficult to prove. In this paper, w...