A study on Type-2 isomorphic circulant graphs. Part 6: Abelian groups $(T{2}_{n,m}(C_n(R)),\circ )$ and $(V_{n,m}(C_n(R)),\circ )$

This study is the $6^{th}$ part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10}. In this part, we define $V_{n,m}(C_n(R))$ and Type-2 set $T{2}_{n,m}(C_n(R))$ of $C_n(R)$ and present their properties. We prove that $(V_{n,m}(C_n(R)),\circ )$ is an Abelian group and $(T{2}_{n,m}(C_n(R)),\circ )$ is a subgroup of $(V_{n,m}(C_n(R)),\circ )$ where $T{2}_{n,m}(C_n(R))$ = $\{C_n(R)\}$ $\cup$ $\{C_n(S):$ $C_n(S)$ is Typ-2 isomorphic to $C_n(R)$ w.r.t. $m\}$ and $(T{2}_{n,m}(C_n(R)),\circ )$ is the Type-2 group of $C_n(R)$ w.r.t. $m$. We also present many examples of Type-1 and Type-2 groups where $T{1}_n(C_n(R))$ = $\{C_n(xR):x\in {\phi }_n\}$ is the Type-1 set of $C_n(R)$ and $(T{1}_n(C_n(R)),{\circ }^{\mathrm{\prime }})$ is its Type-1 group.