  We address the degree-diameter problem for Cayley graphs of Abelian groups
(Abelian graphs), both directed and undirected. The problem turns out to be
closely related to the problem of finding efficient lattice coverings of
Euclidean space by shapes such as octahedra and tetrahedra; we exploit this
relationship in both directions. In particular, we find the largest Abelian
graphs with 2 generators (dimensions) and a given diameter. (The results for 2
generators are not new; they are given in the literature of distributed loop
networks.) We find an undirected Abelian graph with 3 generators and a given
diameter which we conjecture to be as large as possible; for the directed case,
we obtain partial results, which lead to unusual lattice coverings of 3-space.
We discuss the asymptotic behavior of the problem for large numbers of
generators. The graphs obtained here are substantially better than traditional
toroidal meshes, but, in the simpler undirected cases, retain certain desirable
features such as good routing algorithms, easy constructibility, and the
ability to host mesh-connected numerical algorithms without any increase in
communication times.
