A study on Type-2 isomorphic circulant graphs. Part 10: Type-2 isomorphic $C_{n{p}^3}(R)$ w.r.t. $m$ = $p$ and related groups

Abstract

This study is the $10^{th}$ part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10}. In this part, we obtain families of Type-2 isomorphic circulant graphs $C_{n{p}^3}(R)$ w.r.t. $m$ = $p$, and related Abelian groups where $p$ is a prime number and $n\in \mathbb{N}$. In its main theorem, it is proved that for $i$ = 1 to $p$, circulant graphs $C_{n{p}^3}(R_i^{n{p}^3,x+yp})$ are isomorphic of Type-2 w.r.t. $m$ = $p$ and they form Abelian group $(T{2}_{n{p}^3,p}(C_{n{p}^3}(R_i^{n{p}^3,x+yp})),\circ )$ where $T{2}_{n{p}^3,p}(C_{n{p}^3}(R_i^{n{p}^3,x+yp}))$ = $\{\hat{I}{¸}_{n{p}^3,p,jn}(C_{n{p}^3}(R_i^{n{p}^3,x+yp}))$ = $C_{n{p}^3}(R_{i+j}^{n{p}^3,x+yp}):$ $j$ = $0,1,...,p-1$ and $i+j$ in $C_{n{p}^3}(R_{i+j}^{n{p}^3,x+yp})$ is calculated under addition modulo $p\}$, $1\le x\le p-1$, $0\le y\le np-1$, $1\le x+yp\le n{p}^2-1$, $y\in {\mathbb{N}}_0$, $p,n{p}^3-p\in R_i^{n{p}^3,x+yp}$ and $i,n,x\in \mathbb{N}$. And using it, a list of $T{2}_{n{p}^3,p}(C_{n{p}^3}(R_i^{n{p}^3,x+yp}))$, each containing $p$ isomorphic circulant graphs $C_{n{p}^3}(R_i^{n{p}^3,x+yp})$ of Type-2 w.r.t. $m$ = $p$, for $p$ = 3,5,7, $n$ = 1,2 and $y$ = 0 is given in the Annexure and more such families of Type-2 isomorphic circulant graphs are presented in \cite{v24}.

20 pages. This is a modified/corrected version of its previous submission