Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given a finite Abelian group $G$ and a generator subset $A\subset G$ of cardinality two,  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$(G,A)$.  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. #R##N#In this work  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.
['F. Aguiló', 'Alicia Miralles', 'Marisa Zaragozá']
I am writing a research paper on a idea Optimal extensions and quotients of 2--Cayley Digraphs
[{"from": "human", "value": "I am writing a research paper on a idea Optimal extensions and quotients of 2--Cayley Digraphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given a finite Abelian group $G$ and a generator subset $A\\subset G$ of cardinality two,  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$(G,A)$.  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. #R##N#In this work  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."}]