I've been given a question which asks me to "list all non-isomorphic trees with exactly 6 vertices".
Whilst trying to work out how to tell whether a graph is isomorphic (which I still don't understand how to do) I stumbled across the answer to this question, shown in the image below:
However, I'm not sure why this is the answer. In my notes, it says that every graph is isomorphic to its compliment [*] (though I also do not understand why this is). Looking at, for example, the second graph from the left. This graph has a connected compliment. Doesn't this mean that it should be isomorphic, since the mapping from itself to its compliment is an isomorphism?
Also from my notes, I've seen that if $G=(V,E)$ and $G'=(V',E')$ and $|E| \neq |E'|$, then $G$ and $G'$ are not isomorphic. Surely this contradicts [*], since many graphs have a compliment with a different number of edges?
Thanks