Perron eigenvectors and the symmetric transportation polytope

Abstract Let x and y be positive vectors in R n . The set of all n × n nonnegative matrices having x and y T as their right and left Perron eigenvectors is a polyhedral convex cone. A cross section of this cone is the polytope P (x,y) consisting of all n × n nonnegative matrices C such that Cx = x and y T C = y T . The set of doubly stochastic matrices is obtained as a special case when x = y =(1,1,…,1) T . Our purpose is to investigate the structure of P (x,y) and especially its extreme points. This is done by transforming the problem into a symmetric transportation polytope, which contains all n × n nonnegative matrices having the same vector z as their row and column sum vector. Using graph-theoretic methods, we investigate the number of extreme points of this polytope. In particular we study vectors z that yield the maximum number of extreme points.