# Largest Graph with n vertices of Diameter 3

The largest graph of diameter 1 is $$K_n$$. The largest graph of diameter $$\neq$$ 1 has diameter 2 and is $$K_n - e$$ for $$e$$ some edge in the graph $$K_n$$. The construction of the graph of diameter $$\neq$$ 1 is removal of 1 edge from $$K_n$$. I believe if I start from here, there is a valid removal of some edges that will get me to an easily describable graph with diameter 3. How should I make this construction?

I want to take the 2 vertices that are distance 2 from each other in the graph for $$d=2$$ and remove all paths of length 2 between them. I believe this leaves 1 of these vertices connected to only 1 of the other $$n-2$$ vertices and the other connected to all of the $$n-2$$ vertices to obtain the maximal graph. How do I cleanly describe the resulting graph $$G$$ if this is indeed the correct answer?