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In the degree-diameter problem for Abelian Cayley and circulant graphs ofdiameter 2 and arbitrary degree d there is a wide gap between the best lowerand upper bounds valid for all d, being quadratic functions with leadingcoefficient 1/4 and 1/2 respectively. Recent papers have presentedconstructions which increase the coefficient of the lower bound to be at orjust below 3/8, but only for sparse sets of degree d related to primes ofspecific congruence classes. By applying results from number theory theseconstructions can be extended to be valid for every degree above somethreshold, establishing an improved asymptotic lower bound approaching 3/8. Theconstructions use the direct product of the multiplicative and additivesubgroups of a Galois field and a specific coprime cyclic group. Bygeneralising this method an improved upper bound, with quadratic coefficient3/8, is established for this class of construction of Abelian Cayley andcirculant graphs. Analysis of the order of the known extremal diameter 2circulant graphs, up to degree 23, is shown to provide tentative support for aquadratic coefficient of 3/8 for the asymptotic upper bound for the order ofgeneral diameter 2 circulant graphs of arbitrary degree.

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