A study on Type-2 isomorphic circulant graphs. Part 6: Abelian groups $(T2_{n,m}(C_n(R)), \circ)$ and $(V_{n,m}(C_n(R)), \circ)$

TL;DR

This paper defines and analyzes Abelian groups related to Type-2 isomorphic circulant graphs, proving their properties and subgroup relations, and providing examples of Type-1 and Type-2 groups.

Contribution

It introduces the concepts of $V_{n,m}(C_n(R))$ and $T2_{n,m}(C_n(R))$, proving their group structures and subgroup relations within circulant graph theory.

Findings

$(V_{n,m}(C_n(R)), ullet)$ is an Abelian group.

$(T2_{n,m}(C_n(R)), ullet)$ is a subgroup of $(V_{n,m}(C_n(R)), ullet)$.

Examples of Type-1 and Type-2 groups are provided.

Abstract

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 $T2_{n,m}(C_n(R))$ of $C_n(R)$ and present their properties. We prove that $(V_{n,m}(C_n(R)), \circ)$ is an Abelian group and $(T2_{n,m}(C_n(R)), \circ)$ is a subgroup of $(V_{n,m}(C_n(R)), \circ)$ where $T2_{n,m}(C_n(R))$ = $\{C_n(R)\}$ $\cup$ $\{C_n(S):$ $C_n(S)$ is Typ-2 isomorphic to $C_n(R)$ w.r.t. $m \}$ and $(T2_{n,m}(C_n(R)), \circ)$ 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 $T1_{n}(C_n(R))$ = $\{C_n(xR): x\in\varphi_{n}\}$ is the Type-1 set of $C_n(R)$ and $(T1_{n}(C_n(R)), \circ')$ is its Type-1 group.