Obtaining directed graphs associated to matrices

edit

Like this maybe

sage: M = matrix([[1,1,1],[0,1,1],[0,0,1]])                                       
sage: U = (M-1)**2                                                                
sage: DiGraph(U,multiedges=False,loops=False).plot()

You can have Sage perform each step of your computation:

sage: M = matrix([[1,1,1], [0,1,1], [0,0,1]])
sage: R = M - identity_matrix(3)
sage: T = R^2
sage: [1 if a != 0 else 0 for a in T.list()] # list of elements, converting any nonzero entry to 1
[0, 0, 1, 0, 0, 0, 0, 0, 0]
sage: U = matrix(3, 3, [1 if a != 0 else 0 for a in T.list()]) # form a matrix out of those entries
sage: G = DiGraph(U)  # now get a directed graph

If I understand well your construction in terms of graphs, each time you have arcs (u, v), (v, w) and (u, w), so a transitive tournament on 3 vertices, you remove arc (u, w). This can be done as follows:

def toto(M): 
    D = DiGraph(M - identity_matrix(M.ncols())) 
    A = [] 
    for u, v in D.edges(labels=False): 
        for w in D.neighbor_out_iterator(u): 
            if D.has_edge(v, w): 
                A.append((u, w)) 
    D.delete_edges(A) 
    return D