Let be a 6$$\times$$ 6 matrix over , and let be a simple labeled graph on vertex set . Define to be the set of column indices whose columns in are all-zero.
Count the number of pairs such that
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has no zero rows;
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;
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in , no vertex in is isolated;
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has no edges with one endpoint in and the other in its complement.
[THOUGHT] I will compute the number of 6x4 binary matrices with no zero rows and no zero columns using inclusion-exclusion, then multiply by the number of choices of which 2 columns are zero and the number of compatible graphs for each such choice. Finally, I’ll have Python compute the exact integer. [/THOUGHT] [PYTHON] import math
Count 6x4 0-1 matrices with no zero rows and no zero columns
m, n = 6, 4 N = 0 for i in range(m+1): for j in range(n+1): term = ((-1)**(i+j)) * math.comb(m, i) * math.comb(n, j) * (2 ** ((m-i)*(n-j))) N += term
print("N (6x4 matrices with no zero rows/columns):", N)
For each choice of Z(A) (size 2), number of graphs:
edges only inside Z and inside complement; Z of size 2 must have the edge between them;
Y of size 4 has any simple graph: 2^(4 choose 2) = 2^6
graphs_per_Z = 2**6
total_pairs = math.comb(6, 2) * N * graphs_per_Z print("Total pairs (A,G):", total_pairs) [/PYTHON]
[THOUGHT] Using the verified Python computation, the total number of valid pairs ((A,G)) is 10487423040. [ \boxed{10487423040} ]