A study on Type-2 isomorphic circulant graphs. Part 7: Isomorphism series, digraph and graph of $C_n(R)$

Abstract

This study is the $7^{th}$ part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10}. In this study, we define {\em isomorphic set}, {\em isomorphism series}, {\em isomorphism digraph} $\mathcal{D}$ or {\em isomorphism diagram} and {\em isomorphism graph} $\mathcal{G}$ of circulant graphs and obtain these corresponding to $C_{16}(R)$, $C_{27}(S)$ and $C_{54}(1,3,17,19)$ and present the isomorphism digraph and the isomorphism graph of $C_{432}(16, 27, 48, 54, 128, 160, 189)$ which has isomorphic circulant graphs of Type-2 w.r.t. $m$ = 2 as well as $m$ = 3. We also show that each pair of circulant graphs $C_{54}(1,3,17,19)$, $C_{54}(5,13,21,23)$; $C_{54}(7, 11, 21, 25)$, $C_{54}(7, 11, 15, 25)$; and $C_{54}(1,3,17,19)$, $C_{54}(7,11,15,25)$ are isomorphic but they are neither of Type-1 nor of Type-2 w.r.t. $m$ = 3. More such circulant graphs are given in the conclusion. We also define {\em diameter of isomorphic set} of $C_n(R)$ and {\em isomorphic distance} of $C_n(S)$ and $C_n(T)$ and obtained these values for some circulant graphs.