A study on Type-2 isomorphic circulant graphs. Part 2: Type-2 isomorphic circulant graphs of orders 16, 24, 27

TL;DR

This paper extends the study of Type-2 isomorphic circulant graphs, providing new classifications for orders 16, 24, and 27, and calculating the number of isomorphic graph pairs and triples.

Contribution

It introduces a modified definition of Type-2 isomorphism and applies it to classify circulant graphs of specific orders, revealing their isomorphism counts.

Findings

Identified 8 pairs of Type-2 isomorphic graphs of order 16.

Found 32 pairs of Type-2 isomorphic graphs of order 24.

Discovered 12 triples of Type-2 isomorphic graphs of order 27.

Abstract

This study is the $2^{nd}$ part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10}. Definition of Type-2 isomorphism of circulant graphs $C_n(R)$ w.r.t. $m$ was further modified by the author by considering $m > 1$ divides $\gcd(n, r)$, $m^3$ divides $n$ and $r\in R$ and studied Type-2 isomorphic circulant graphs w.r.t. $m$ = 2. This modification simplifies our calculations while finding isomorphic circulant graphs of Type-2. In this paper, using the modified definition \ref{d4.2}, we obtain Type-2 isomorphic circulant graphs of orders 16, 24 and 27 and show that the total number of pairs of Type-2 isomorphic circulant graphs of orders 16 and 24 are 8 and 32, respectively and the total number of triples of Type-2 isomorphic circulant graphs of order 27 are 12.