Wilson Peraprakash | Searcharxiv

arXiv2026

A study on Type-2 isomorphic circulant graphs. PART 9: Computer programs to show Type-1 $\&$ -2 isomorphic circulant graphs

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 of $C_n(R)$ w.r.t. $m$ $\ni$ $m$ = $\gcd(n, r) > 1$, $r\in R$ and $r,n\in\mathbb{N}$ and studied such graphs for $m$ = 2 in \cite{v13,v20}. But obtaining Type-2 isomorphic circulant graphs is not easy. Using a $C^{++}$ computer program, the authors obtained families of Type-2 isomorphic $C_{n}(R)$ w.r.t. $m$ = 2,3,5,7 for $n\in\mathbb{N}$ as well as $C_{np^3}(R)$ w.r.t. $m$ = $p$ for $n\in\mathbb{N}$ and $p$ is an odd prime. In this paper, we present the $C^{++}$ program and also a VB program POLY415.EXE which is used to show how Type-1 and Type-2 isomorphisms of a circulant graph take place as well as for checking and finding Type-1 and Type-2 circulant graphs of a given order and is very useful to develop its theory on Type-2 isomorphic circulant graphs \cite{v2-1}-\cite{v2-10}.

math.CO↗

arXiv2022

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}.

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