A study on Type-2 isomorphic circulant graphs. Part 1: Type-2 isomorphic circulant graphs $C_n(R)$ w.r.t. $m$ = 2

This study is the first part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10}. Circulant graphs $C_n(R)$ and $C_n(S)$ are said to be \emph{Adam’s isomorphic} if there exist some $a\in {\mathbb{Z}}_n^{\ast }$ such that $S=aR$ under arithmetic reflexive modulo $n$ \cite{ad67}. In this paper, the author modified his earlier definition \cite{v96} of Type-2 isomorphism w.r.t. $m$ such that $m$ and $m^3$ are divisors of $\gcd (n,r)$ and $n$, respectively, and $r\in R$. Using the modified definition, we present our study on Type-2 isomorphism of circulant graphs $C_n(R)$ w.r.t. $m$ = 2. We prove that $(i)$ $C_{16}(1,2,7)$ and $C_{16}(2,3,5)$ are Type-2 isomorphic w.r.t. $m$ = 2; $(ii)$ For $n\ge 2$, $k\ge 3$, $1\le 2s-1\le 2n-1$, $n\mathrm{\ne }2s-1$, $R$ = $\{2,2s-1,4n-(2s-1)\}$ and $S$ = $\{2,2n-(2s-1),2n+2s-1\}$, $C_{8n}(R)$ and $C_{8n}(S)$ are Type-2 isomorphic w.r.t. $m$ = 2, $n,s\in \mathbb{N}$; and $(iii)$ For $n\ge 2$, $1\le 2s-1<2s^{\mathrm{\prime }}-1\le [\frac{n}{2}]$, $0\le t\le [\frac{n}{2}]$, $R$ = $\{2,2s-1,2s^{\mathrm{\prime }}-1\}$ and $n,s,s^{\mathrm{\prime }}\in \mathbb{N}$, if ${\theta }_{n,2,t}(C_n(R))$ and $C_n(R)$ are isomorphic circulant graphs of Type-2 w.r.t. $m$ = 2 for some $t$, then $n\equiv 0(mod8)$, $2s-1+2s^{\mathrm{\prime }}-1$ = $\frac{n}{2}$, $2s-1\mathrm{\ne }\frac{n}{8}$, $t$ = $\frac{n}{8}$ or $\frac{3n}{8}$, $1\le 2s-1\le \frac{n}{4}$ and $n\ge 16$ where ${\theta }_{n,m,t}$ is a transformation used to define Type-2 isomorphism of a circulant graph. At the end, we present a VB program POLY215.EXE which shows how Type-2 isomorphism w.r.t. $m$ = 2 of $C_{8n}(R)$ takes place for $R=\{2,2s-1,4n-(2s-1)\}$, $n\ge 2$ and $n,s\in \mathbb{N}$.