On abelian cayley graphs of diameter two and defect one

For an abelian Cayley graph of degree 2nand diameter 2, it has at most 2n(2)+2n+1 vertices which meets the abelian Cayley-Moore bound. Recently, Leung and thesecond author proved that such a graph exists if and only if n=1,2. A naturalquestion is that whether one can also classify abelian Cayley graphs of degree 2nanddiameter 2 with exactly 2(n)2+2nvertices. For n=1,2, there are examples. As thetotal number of vertices of such graphs is one smaller than the abelian Cayley-Moorebound, we call them of defect one. By using some algebraic approaches, we provideseveral nonexistent results for in?nitely many n>2.