Is there an infinite class of graphs with diameter 2, girth 5, and minimum degree at least 2?
Girth 5 is necessary, since otherwise complete bipartite graphs are an answer. Minimum degree at least 2 is necessary, since otherwise stars are an answer.
The Hoffman–Singleton graph (which has 60 vertices) is the largest graph I know with the properties in the question.
The question is interesting if "minimum degree at least 2" is replaced by "minimum degree at least 3", but I think it makes little difference because degree 2 vertices are highly restricted in a graph with diameter 2.
This question is related (but different) to the question Girth and diameter of a graph with minimum degree at least 3.