Row-stochastic matrices with a common left fixed vector

AbstractWe consider the convex polytope Sn(x) that consist of those n×n (row) stochastic matrices having a common nonnegative (left) fixed vector xt. We examine the 1-skeleton of Sn(x) and show how to construct all extreme points adjacent to a given one (as vertices of the 1-skeleton). Connections with transportation polytopes are discussed. Further, we give a formula for the degree of an extreme point in the 1-skeleton of Sn(x), find its maximum and minimum values, and determine when all degrees are equal. An explicit description of the 1-skeleton is given for n=3