# Find all graphs with valency sequence $(4, 2, 2, 2, 1, 1)$

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? – Brian M. Scott May 31 '12 at 11:34
What following sequences? Can you do something so the title and the body agree, and make sense? – Gerry Myerson May 31 '12 at 11:35
Perhaps you should start by considering the vertex of degree 4. Draw it and the 4 neighbors. What can you do next? – John Engbers May 31 '12 at 11:38

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:

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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: faculty.capebretonu.ca/jpreen/410112w1.mws – jp26 Jun 1 '12 at 15:50
@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. – Listing Jun 1 '12 at 20:14