Strong linear preservers of symmetric doubly stochastic or doubly substochastic matrices

Let denote either the set of n×n symmetric doubly stochastic matrices or the set of n×n symmetric doubly substochastic matrices and let T be a linear map on span . We prove that if and only if there exists an n×n permutation matrix P such that T(X)=PtXP for all . Our proofs make use of the concept of neighborly extreme points of a polytope and depend on some intricate graph theory.