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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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    The title and the question don’t agree; are you interested in finding **one** such graph, or in finding **all** of them?2012-05-31
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    What following sequences? Can you do something so the title and the body agree, and make sense?2012-05-31
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    Perhaps you should start by considering the vertex of degree 4. Draw it and the 4 neighbors. What can you do next?2012-05-31

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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:

enter image description here

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    Thanks for finding that page that I created back in 2003, I forgot I ever did it! This year I was teaching my course again and this time I made up a Maple worksheet (using the GraphTheory package which is available since Maple 13). It makes a list of all simple graphs with any given valency sequence, and is available here: http://faculty.capebretonu.ca/jpreen/410112w1.mws2012-06-01
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    @jp26 neat to see that you are using stackexchange too. I found your description of the algorithm very straight forward and easy to understand, so I linked it.2012-06-01