Let $G$ be a simple graph on $n$ vertices $(n > 3)$ with no vertex of degree $n − 1$. Suppose that for any two vertices of $G$, there is a unique vertex adjacent to both of them. It is an exercise from A Course in Combinatorics, van Lint and Wilson, to prove that such a graph is regular.
My question, however, is whether graphs satisfying the given constraints even exist. While discussing the original exercise during a problem-solving session, someone asked if we could come up with an example of a graph where every pair of vertices have an unique common neighbor, and there are no global vertices. Neither were we able to come up with a concrete example or procedure for construction, nor did we establish a proof that no graph has these properties.
Note: as for proving that such a graph is regular, it turns out to be fairly straightforward, the rough idea is to pair off the neighbors of every pair of vertices using the unique-common-neighbor criteria to establish the fact that every pair of vertices have the same degree, and then a transitivity argument, with the help of the no-global-vertex constraint, gives us that the graph is regular.