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

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 $C_{np^3}(R)$ 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 $C_{np^3}(R^{np^3,x+yp}_i)$ are isomorphic of Type-2 w.r.t. $m$ = $p$ and they form Abelian group $(T2_{np^3,p}(C_{np^3}(R^{np^3,x+yp}_i)), \circ)$ where $T2_{np^3,p}(C_{np^3}(R^{np^3,x+yp}_i))$ = $\{\theta_{np^3,p,jn}(C_{np^3}(R^{np^3,x+yp}_i))$ = $C_{np^3}(R^{np^3,x+yp}_{i+j}) :$ $j$ = $0,1,...,p-1$ and $i+j$ in $C_{np^3}(R^{np^3,x+yp}_{i+j})$ is calculated under addition modulo $p \}$, $1 \leq x \leq p-1$, $0 \leq y \leq np - 1$, $1 \leq x+yp \leq np^2-1$, $y\in\mathbb{N}_0$, $p,np^3-p\in R^{np^3,x+yp}_i$ and $i,n,x\in\mathbb{N}$. And using it, a list of $T2_{np^3,p}(C_{np^3}(R^{np^3,x+yp}_i))$, each containing $p$ isomorphic circulant graphs $C_{np^3}(R^{np^3,x+yp}_i)$ 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}.