Describe the edges of bn in terms of the permutations

Question

Math 121c: Topics in Geometric Combinatorics, Spring 2012 Problems-

Consider the polytope B_n ⊂ R^n×n consisting of n × n matrices [a_i,j ]_1≤i,j≤n satisfying the following equations and inequalities:

_i=1∑ⁿ a_i,j = _j=1∑ⁿa_i,j = 1, and  a_i,j ≥ 0, for all 1 ≤ i, j ≤ n.

(a) Prove that B_n has dimension (n - 1)².

(b) Prove that the vertices of B_n are the n × n permutation matrices.

(c) Describe the edges of B_n in terms of the permutations associated to its vertices. Conclude that the graph B_n has diameter 2.

(d) Determine the facets of B_n.