Number of possible graphs from a reachability matrix?

Question

I need to know how to work out how many possible different digraphs can be drawn from a given reachability matrix. It needs to be with the minimum number of arcs between the nodes within the graph (which i believe is 10 for the one given). If someone could explain it step by step for this reachability matrix i would be grateful. \begin{pmatrix} 1&0&0&1&0&0&0&0&0 \\ 1&1&1&1&1&1&1&1&0 \\ 1&0&1&1&1&1&1&1&0 \\ 1&0&0&1&0&0&0&0&0 \\ 1&0&1&1&1&1&1&1&0 \\ 1&0&0&1&0&1&0&0&0 \\ 1&0&1&1&1&1&1&1&0 \\ 1&0&1&1&1&1&1&1&0 \\ 1&0&1&1&1&1&1&1&1 \end{pmatrix}

Answer 1

So I am pretty sure I have worked out the answer to my own question, if anyone would like to confirm or disprove my working for it in the comments then I welcome your input.

There are 4 interchangeable nodes, 3, 5, 7 and 8, which gives 4! worth of different graphs to begin with. Then there are another 2 interchangeable nodes, 1 and 4, which is 2! MORE graphs. So far we have 4! * 2! which gives 24 * 2 so 48 graphs. Then nodes 2 and 9 can be placed next to 4 different nodes each without affecting the matrix, they can be connected to 3, 5, 7 or 8. This gives a multiple of 16 (4 possible connections * another 4 possible connections) MORE graphs. Thus giving an answer of 48 * 16 or 768 possible graphs with the minimum number of connections between the nodes.