Improved upper bounds for the order of some classes of Abelian Cayley and circulant graphs of diameter two

In the degree-diameter problem for Abelian Cayley and circulant graphs of diameter 2 and arbitrary degree d there is a wide gap between the best lower and upper bounds valid for all d, being quadratic functions with leading coefficient 1/4 and 1/2 respectively. Recent papers have presented constructions which increase the coefficient of the lower bound to be at or just below 3/8, but only for sparse sets of degree d related to primes of specific congruence classes. By applying results from number theory these constructions can be extended to be valid for every degree above some threshold, establishing an improved asymptotic lower bound approaching 3/8. The constructions use the direct product of the multiplicative and additive subgroups of a Galois field and a specific coprime cyclic group. By generalising this method an improved upper bound, with quadratic coefficient 3/8, is established for this class of construction of Abelian Cayley and circulant graphs. Analysis of the order of the known extremal diameter 2 circulant graphs, up to degree 23, is shown to provide tentative support for a quadratic coefficient of 3/8 for the asymptotic upper bound for the order of general diameter 2 circulant graphs of arbitrary degree.