# Obtaining directed graphs associated to matrices

Let $M$ be an $n \times n$-matrix with entries only 0 or 1 and all diagonal entries equal to 1. (usually $M$ is upper triangular) Let $R$ be the same matrix as $M$ but with all diagonal entries set to zero. Let $U=(u_{i,j})$ be the matrix with 1 as an entry if $R^2$ (the matrix product of $R$ with itself) has a non-zero entry in the same position and let $U$ have 0 in this entry if $R^2$ has 0 as an entry in this position. Let $C=(c_{i,j})$ be the matrix $R-U$. Let $G_M$ be the directed graph with $n$ vertices and there is an arrow from $i$ to $j$ if and only if $c_{i,j}$ is 1.

For example when $M$ is the matrix with rows $[1,1,1],[0,1,1],[0,0,1]$ then the graph $G_M$ has 3 vertices with an arrow from 1 to 2 and an arrow from 2 to 3.

My question is whether there is a quick method to obtain all such 0-1 matrices with Sage for a given $n$ and the associated graph $G_M$ displayed as a picture (and as a graph in sage).

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In your example, R^2 will have a single nonzero entry, in entry (1, 3). So why does $G_M$ have two arrows?

( 2021-03-30 18:18:29 +0100 )edit

Thank you for your comment. I forgot that the graph is defined via the matrix $R-U$ instead of $U$. I hope it it correct now.

( 2021-03-30 19:41:51 +0100 )edit

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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()

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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

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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

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