Optimal extensions and quotients of 2--Cayley Digraphs

TL;DR

This paper introduces procedures to generate optimal extensions and quotients of 2-Cayley digraphs, leading to new optimal families and improved diameters in the context of finite Abelian groups.

Contribution

The paper defines procedures E and Q for constructing optimal extensions and quotients of 2-Cayley digraphs, and proves their properties, including optimality preservation.

Findings

Quotients obtained by procedure Q are also optimal.

Characterization of the number of tight extensions generated by procedure E.

Identification of conditions for infinite tight extensions.

Abstract

Given a finite Abelian group $G$ and a generator subset $A\subset G$ of cardinality two, we consider the Cayley digraph $\Gamma=$Cay$(G,A)$. This digraph is called $2$--Cayley digraph. An extension of $\Gamma$ is a $2$--Cayley digraph, $\Gamma'=$Cay$(G',A)$ with $G<G'$, such that there is some subgroup $H<G'$ satisfying the digraph isomorphism Cay$(G'/H,A)\cong$Cay$(G,A)$. We also call the digraph $\Gamma$ a quotient of $\Gamma'$. 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.