What are the triangle free graphs on $\lfloor\frac{n^2}{4}\rfloor$ edges .

Question

I tried for $n=4$, it is a cycle of length $4$ which is $K_{2,2}$. $n=5$, it seems to be $K_{2,3}$. So my guess is it is a complete bipartite graph?

Answer 1

This is a partial case of Turán’s theorem for $r=2$. The unique (up to a isomorphism) edge-maximal triangle-free graph is a Turán graph, which is a complete bipartite graph on $n$ vertices such that the sizes of the parts differs by at most one.