# Find as many non-isomorphic self-complementary graphs as possible (with up to $7$ vertices)

Find as many non-isomorphic self-complementary graphs as possible with up to $7$ vertices. State why there aren't more.

I've checked what a self-complementary graph is and wikipedia is saying: "A self-complementary graph is a graph which is isomorphic to its complement."

So then what is a non-isomorphic self-complementary graph? A graph which is non-isomorphic to its complement?

Here are some I found but I'm not sure if this is correct. All in all, this sounds very confusing, contradicting. Or did I just understand it wrong? How would you find more graphs? Editing my question for Henning Makholm's answer: • A graph can't be "non-isomorphic" by itself, just like a positive integer can't be "coprime" by itself. Perhaps it would be clearer if the question said "find as many self-complementary graphs as possible with up to 7 vertices, up to isomorphism". – NoName Nov 18 '17 at 23:25

It will help your search to notice that $K_6$ and $K_7$ both have an odd number of edges, so there is no self-complementary graph on $6$ or $7$ vertices. Having $2$ or $3$ vertices is taken care of by the same argument, and you have already found the single example (up to isomorphism) on $4$ vertices.
What is left for you is then only to enumerate graphs with $5$ vertices and $5$ edges, and check which of them are self-complementary.
• @cnmesr: Yes, that is one. There is at least one more with $5$ vertices, in addition to the trivial graph. – Henning Makholm Nov 18 '17 at 20:50
• Thank you! Do you know what exactly is the reason there aren't more of these kind of graphs (with more than $7$ vertices)? – cnmesr Nov 18 '17 at 22:58