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

Vilfred Kamalappan

paper

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

arXiv:2605.12868

Abstract

This study is the $6^{t h}$ 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 $T 1_{n} (C_{n} (R))$ = ${C_{n} (x R) : x \in I ¨ _{n}}$ is the Type-1 set of $C_{n} (R)$ and $(T 1_{n} (C_{n} (R)), \circ^{'})$ is its Type-1 group.

arXiv admin note: substantial text overlap with arXiv:2605.11441

A study on Type-2 isomorphic circulant graphs. Part 6: Abelian groups (T2n,m​(Cn​(R)),∘) and (Vn,m​(Cn​(R)),∘)

Abstract

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