论文标题
2型同构循环图$ np^3 $ W.R.T.的家族$ r = p $和他们的阿贝尔团体
Families of Type-2 Isomorphic Circulant Graphs of Order $np^3$ w.r.t. $r = p$ and Their 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 $。 $ c_n（r）$据说具有{\ it cayley同构}（ci） - property，如果每当$ c_n（s）$是同构对$ c_n（r）时，它们是亚当的同构。 CI问题确定哪些图（或哪些组）具有$ ci $ - property。循环$ CI $ -Groups的分类已经完成，但是对没有$ CI $ - 杂制的图表进行了调查。 VILFRED定义的类型2同构，与Adam的同构，循环图$ C_N（R）$ W.R.T. $ r $，$ r \ in r $ $ \ ni $ $ \ gcd（n，r）= m> 1 $。类型-2同构循环图没有Ci-Property，我们获得了此类订单$ n $的图形，$ r $ = 2,3,5,7，$ n \ in \ Mathbb {n} $。在本文中，我们获得了订单$ NP^3 $ W.R.T.的类型2同构循环图。 $ r = p $，以及这些同构图上的阿贝利安组，其中$ p $是质量数字，而$ n \ in \ mathbb {n} $中的$ n \。定理\ ref {c10}和\ ref {c13}是主要结果。使用Theorem \ ref {C13}，$ p $同构循环图上的Abelian组列表$ C_ {np^3}（r^{np^3，x+yp} _i）$ 2 type-2 w.r.r.t.类型$ r = p $ for $ i $ = 1至$ p $，对于$ p $ = 3,5,7，$ n $ = 1,2和$ y $ = 0，在附件中给出，$ 1 \ leq x \ leq x \ leq p -1 $，$ y \ in \ mathbb {n} np^2-1 $，$ p，np^3-p \ in r^{np^3，x+yp} _i $和$ i，n，x \ in \ mathbb {n} $。
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 $n$. $C_n(R)$ is said to have {\it Cayley Isomorphism} (CI)-property if whenever $C_n(S)$ is isomorphic to $C_n(R),$ they are of Adam's isomorphic. CI-problem determines which graphs (or which groups) have the $CI$-property. Classification of cyclic $CI$-groups was completed but investigation of graphs without $CI$-property is not much done. Vilfred defined Type-2 isomorphism, different from Adam's isomorphism, of circulant graphs $C_n(R)$ w.r.t. $r$, $r\in R$ $\ni$ $\gcd(n, r) = m > 1$. Type-2 isomorphic circulant graphs don't have CI-property and we obtained such graphs of order $n$ for $r$ = 2,3,5,7, $n\in\mathbb{N}$. In this paper, we obtain Type-2 isomorphic circulant graphs of order $np^3$ w.r.t. $r = p$, and abelian groups on these isomorphic graphs where $p$ is a prime number and $n\in\mathbb{N}$. Theorems \ref{c10} and \ref{c13} are the main results. Using Theorem \ref{c13}, a list of abelian groups on the $p$ isomorphic circulant graphs $C_{np^3}(R^{np^3,x+yp}_i)$ of Type-2 w.r.t. $r = p$ for $i$ = 1 to $p$ and for $p$ = 3,5,7, $n$ = 1,2 and $y$ = 0 is given in the Annexure, $1 \leq x \leq p-1$, $y\in\mathbb{N}_0$, $0 \leq y \leq np - 1$, $1 \leq x+yp \leq np^2-1$, $p,np^3-p\in R^{np^3,x+yp}_i$ and $i,n,x\in\mathbb{N}$.
