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 for sparse sets of degree d related to primes of specific congruence classes. The constructions use the direct product of the multiplicative and additive subgroups of a Galois field and a specific cyclic group of coprime order. It was anticipated that this approach would be capable of yielding further improvement towards the upper bound value of 1/2. In this paper, however, it is proved that the quadratic coefficient of the order of families of Abelian Cayley graphs of this class of construction can never exceed the value of 3/8, establishing an asymptotic limit of 3/8 for the quadratic coefficient of families of extremal graphs of this class. By applying recent 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 for general Abelian Cayley and circulant graphs of diameter 2 and arbitrary degree d.