A study on Type-2 isomorphic circulant graphs. Part 8: Isomorphism series, digraph and graph of $C_n(R)$

Abstract

This article is the $8^{th}$ 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} or {\em isomorphism diagram} $\mathcal{D}$ 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 $C_{432}(T)$ which has isomorphic circulant graphs of Type-2 w.r.t. $m$ = 2,3 and of $C_{6750}(U)$ which has isomorphic circulant graphs of Type-2 w.r.t. $m$ = 3,5 where $T$ = $\{16,27,48,54,128,160,189\}$ and $U$ = $\{135,243,250,750,1107,1593,2000,2457,2500,2943\}$. We also define {\em diameter of isomorphic set} of $C_n(R)$ and {\em isomorphic distance} between any two circulant graphs. We find that the order of isomorphic sets of $C_{432}(T)$ and $C_{6750}(U)$ are 36 and 450 and their diameters are 3 and 3, respectively.

arXiv commentThis article contains 79 pages. Previous version of Part 7 and Part 8 are interchanged, modified and developed to this stage