Abstract :
A nonnegative matrix T = (tij)i,j=tn is a generalized transitive tournament matrix (GTT matrix) if tjj = 0, tij = 1 −tji for i ≠ j, and 1 ⩽ tij + tjk + tki ⩽ 2 for i,j,k pairwise distinct. An approach to the problem of characterize the set of vertices of the polytope GTTn of all GTT matrices of order n was the introduction by Brualdi and Hwang of the ∗-graph associated to each T ϵ GTTn. We introduce a new graph which generalize the ∗-graph. The new graph will be employed to develop a computable criterion for determine whether any given GTT matrix of order n is or not a vertex of GTTn. A consequence of the criterion is that if T is a vertex of GTTn. with small number, r, of different entries then we have strong restrictions for the possible entries of T. Namely, if r ⩽ 6 then the set of entries of T is equal to 0, 1, 0, 1, 0,12,1,0,13,23,1, 0,13,12,23,1, 0,14,12, 34,1, 0,16,13,23,56,1, or 15,25,35,45,1.
