Optimal extensions and quotients of 2-Cayley Digraphs

Given a finite Abelian group G and a generator subset A ⊂ G of cardinality two, we consider the Cayley digraph Γ = Cay(G, A). This digraph is called 2–Cayley digraph. An extension of Γ is a 2–Cayley digraph, Γ′ = Cay(G′, A) with G < G′, such that there is some subgroup H < G′ satisfying the digraph isomorphism Cay(G′/H, A) ∼= Cay(G, A). We also call the digraph Γ a quotient of Γ′. Notice that the generator set does not change. A 2–Cayley digraph is called optimal when its diameter is optimal with respect to its order. In this work we define two procedures, E and Q, which generate a particular type of extensions and quotients of 2–Cayley digraphs, respectively. These procedures are used to obtain optimal quotients and extensions. Quotients obtained by procedure Q of optimal 2–Cayley digraphs are proved to be also optimal. The number of tight extensions, generated by procedure E from a given tight digraph, is characterized. Tight digraphs for which procedure E gives infinite tight extensions are also characterized. Finally, these two procedures allow the obtention of new optimal families of 2–Cayley digraphs and also the improvement of the diameter of many proposals in the literature.