Families of Circulant Graphs Without CI-Property and More Abelian Groups

V. Vilfred Kamalappan

Published 2020-12-18Version 1

A circulant graph $C_n(R)$ is said to have the Cayley Isomorphism (CI) property if whenever $C_n(S)$ is isomorphic to $C_n(R),$ there is some $a\in \mathbb{Z}_n^*$ for which $S = aR$. In this paper, we obtain many families of Type-2 isomorphic circulant graphs and new abelian groups. Type-2 isomorphism of circulant graphs is a new type of isomorphism, different from already known Adam's or Type-1 isomorphism and Type-2 isomorphic circulant graphs have the property that they are without $CI$-property. The main results are Theorems \ref{d4} and \ref{d7}. Using Theorem \ref{d7} and Lemma \ref{d9}, a list of new abelian groups, $(T2_{np^3,p}(C_{np^3}(R^{np^3,x+yp}_i)),~\circ)$ are given in the annexure for $n$ = 1 to 5, $p$ = 3,5,7, $x$ = 1 to $p-1$, $y$ = 0 to $np-1$ and $p,np^3-p\in R^{np^3,x+yp}_i$ and for $p$ = 11, $n$ = 1 to 2, $x$ = 1 to $p-1$, $y$ = 0 to $np-1$ and $p,np^3-p\in R^{np^3,x+yp}_i$. For more clarity, a list, without using Lemma \ref{d9}, is given in \cite{vw0A} for $n$ = 1 to 5, $p$ = 3,5,7,11, $x$ = 1 to $p-1$, $y$ = 0 to $np-1$ and $p,np^3-p\in R^{np^3,x+yp}_i$.