Test the following sequences using to see if they are graphical or not: (if they are graphical draw an exemplar graph having that valency sequence)
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There is an algorithm to easily verify valency sequences.
Enumering all possible graphs is a slightly more difficult problem. As there are only a few graphs with 6 nodes you can however figure out that there are only 2 possible graphs with that sequence: (you can modify the algorithm to give you all possible graphs)
Graphs 1 and 2: