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A binary matrix is a real matrix with entries 0 and 1.
Comments from Brendan McKay (bdm(AT)cs.anu.edu.au), Aug 21 2006: Equivalently, directed graphs (simple but loops allowed) without a few small forbidden subgraphs (those allowing 2 distinct paths of length 2 from vertex x to vertex y for some x,y; I think there are 6 possibilities). One can also consider isomorphism classes of those digraphs.
Comment Rob Pratt (Rob.Pratt(AT)sas.com), Aug 03 2008: A121294 provides a lower bound on the maximum number of 1's in such a matrix M. There are cases where a higher number is reached; the following 5 X 5 matrix has 11 ones and its square is binary:
0 0 1 0 0
0 0 0 0 1
1 1 0 0 1
1 1 0 1 0
1 1 0 1 0.
The optimal values seem to match A070214, verified for n<=7.
Term (5,1) of n-th power of the 5x5 matrix shown = A001045(n), the Jacobsthal sequence. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 03 2008]
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