A study on Type-2 isomorphic circulant graphs and related Abelian groups

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 $n$. In 1970, Elspas and Turner \cite{eltu} raised a question on the isomorphism of $C_{16}(1,3,7)$ and $C_{16}(2,3,5)$ and Vilfred \cite{v96} gave its answer by defining Type-2 isomorphism, different from Adam’s isomorphism or Type-1 isomorphism, of $C_n(R)$ w.r.t. $m$ where $m

1$isadivisorof$\gcd(n, r)$and$r\in R$. This paper is an extensive study on Type-2 isomorphic circulant graphs. Vilfred and Wilson \cite{vw0A} obtain isomorphic circulant graphs $C_{n{p}^3}(R)$ of Type-2 w.r.t. $m$ = $p$, and related Abelian groups where $p$ is a prime number and $n\in \mathbb{N}$. Using Theorem \ref{c13}, a list of $T{2}_{n{p}^3,p}(C_{n{p}^3}(R_i^{n{p}^3,x+yp}))$ = $\{C_{n{p}^3}(R_j^{n{p}^3,x+yp}):j=1,2,\mathrm{.}\mathrm{.}\mathrm{.},p\}$ for $p$ = 3,5,7,11 and $n$ = 1 to 5 and also for $p$ = 13 and $n$ = 1 to 3 are given in the Annexure where $(T{2}_{n{p}^3,p}(C_{n{p}^3}(R_i^{n{p}^3,x+yp})),\circ )$ is an abelian group on the $p$ isomorphic circulant graphs $C_{n{p}^3}(R_i^{n{p}^3,x+yp})$ of Type-2 w.r.t. $m$ = $p$, $1\le i,j\le p$, $1\le x\le p-1$, $y\in {\mathbb{N}}_0$, $0 \leq y \leq np - 1$,$1 \leq x+yp \leq np^2-1$,$p,np^3-p\in R^{np^3,x+yp}_i$and$i,j,n,x\in$\mathbb{N}$$. We also show existence of isomorphic circulant graphs $C_n(R)$ and $C_n(S)$ which are neither Type-1 nor Type-2 w.r.t. any particular $m$. We use VB program to develop this theory and for illustration of examples.