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Questions about the Tutte polynomial of graphs and matroids, which is a polynomial in two variables encoding many interesting combinatorial informations.
5
votes
Tutte polynomials, graph complements and degree sequences
If you allow $n=8$, the answer to question 1 2 is yes.
The graph on $8$ vertices with edges:
[(0, 3), (0, 5), (0, 6), (0, 7), (1, 4), (1, 6), (1, 7), (2, 4), (2, 6), (3, 5), (3, 7), (4, 5), (4, 7), …