论文标题
一项关于2型同构循环图和相关的阿贝尔组的研究
A study on Type-2 isomorphic circulant graphs and related Abelian groups
论文作者
论文摘要
循环图$ c_n（r）$和$ c_n（s）$据说为\ emph {adam的同构}，如果存在一些$ a \ in \ mathbb {z} _n _n^*$，这样，$ s = a arithmetic反射型Modulo nodulo $ n $。 1970年，Elspas和Turner \ cite {eltu}提出了一个问题，就$ c_ {16}（1，3，7）$和$ c_ {16}（2，3，3，5）$和vilfred \ cite \ cite \ cite {v96}的答案通过定义类型-2 Isomorphism，与indam的indamsimpliss和indams的答案提出了一个问题。 $ C_N（R）$ W.R.T. $ m $其中$ m> 1 $是$ \ gcd（n，r）$和$ r \ in r $的除数。本文是一项关于2型同构循环图的广泛研究。 Vilfred和Wilson \ cite {VW0A}获得ISomorphic循环图$ C_ {NP^3}（r）type-2-2 W.R.T.的$ $ m $ = $ P $，以及相关的Abelian组，其中$ p $是质量数字，$ n \ in \ mathbb {n} $。使用theorem \ ref {c13}，$ t2_ {np^3，p}的列表（c_ {np^3}（r^{np^3，x+yp} _i））$ = $ = $ \ \ { 1,2，...，p $ = 3,5,7,11和$ n $ = 1至5，以及$ p $ = 13和$ n $ = 1至3 $ = 1至3，其中$（t2_ {np^3，p}，p}，p}（c_ {c_ {np^3}（np^3}（np^3}（np^3}（r^{np^3，crimian con）， $ p $ isomorphic循环图$ c_ {np^3}（r^{np^3，x+yp} _i）$ 2 w.r.t. type-2 w.r.t. $ m $ = $ p $，$ 1 \ leq i，j \ leq p $，$ 1 \ leq x \ leq p-1 $，$ y \ in \ mathbb {n} _0 $，$ 0 \ leq y \ leq y \ leq yq np-1 $，$ 1 $，$ 1 \ leq x+yq x+yp x+yp \ yp \ leq np $ $ $ r^{np^3，x+yp} _i $和$ i，j，n，x \ in \ mathbb {n} $。我们还显示了同构循环图的存在$ c_n（r）$和$ c_n（s）$既不是1 type-1也不是2型W.R.T.任何特定的$ m $。我们使用VB程序来开发这一理论并说明示例。
Circulant graphs $C_n(R)$ and $C_n(S)$ are said to be \emph{Adam's isomorphic} if there exist some $a\in \mathbb{Z}_n^*$ such that $S = a R$ 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 $m > 1$ is a divisor of $\gcd(n, r)$ 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 $C_{np^3}(R)$ 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 $T2_{np^3,p}(C_{np^3}(R^{np^3,x+yp}_i))$ = $\{C_{np^3}(R^{np^3,x+yp}_{j}) : j = 1,2,...,p\}$ 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 $(T2_{np^3,p}(C_{np^3}(R^{np^3,x+yp}_i)), \circ)$ is an abelian group on the $p$ isomorphic circulant graphs $C_{np^3}(R^{np^3,x+yp}_i)$ of Type-2 w.r.t. $m$ = $p$, $1 \leq i,j \leq p$, $1 \leq x \leq p-1$, $y\in\mathbb{N}_0$, $0 \leq y \leq np - 1$, $1 \leq x+yp \leq np^2-1$, $p,np^3-p\in R^{np^3,x+yp}_i$ 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.
