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arxiv: 2012.11372 · v11 · pith:6L3YAZMY · submitted 2020-12-18 · math.CO

A study on Type-2 isomorphic circulant graphs and related Abelian groups

V. Vilfred Kamalappan This is my paper

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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 $n$. In 1970, Elspas and Turner \cite{eltu} raised a question on the isomorphism of $C_{16}(1, 3, 7)$ and $C_{16}(2, 3, 5)$ and Vilfred \cite{v96} gave its answer by defining Type-2 isomorphism, different from Adam's isomorphism or Type-1 isomorphism, of $C_n(R)$ w.r.t. $m$ where $m > 1$ is a divisor of $\gcd(n, r)$ and $r\in R$. This paper is an extensive study on Type-2 isomorphic circulant graphs. Vilfred and Wilson \cite{vw0A} obtain isomorphic circulant graphs $C_{np^3}(R)$ of Type-2 w.r.t. $m$ = $p$, and related Abelian groups where $p$ is a prime number and $n\in\mathbb{N}$. Using Theorem \ref{c13}, a list of $T2_{np^3,p}(C_{np^3}(R^{np^3,x+yp}_i))$ = $\{C_{np^3}(R^{np^3,x+yp}_{j}) : j = 1,2,...,p\}$ for $p$ = 3,5,7,11 and $n$ = 1 to 5 and also for $p$ = 13 and $n$ = 1 to 3 are given in the Annexure where $(T2_{np^3,p}(C_{np^3}(R^{np^3,x+yp}_i)), \circ)$ is an abelian group on the $p$ isomorphic circulant graphs $C_{np^3}(R^{np^3,x+yp}_i)$ of Type-2 w.r.t. $m$ = $p$, $1 \leq i,j \leq p$, $1 \leq x \leq p-1$, $y\in\mathbb{N}_0$, $0 \leq y \leq np - 1$, $1 \leq x+yp \leq np^2-1$, $p,np^3-p\in R^{np^3,x+yp}_i$ and $i,j,n,x\in\mathbb{N}$. We also show existence of isomorphic circulant graphs $C_n(R)$ and $C_n(S)$ which are neither Type-1 nor Type-2 w.r.t. any particular $m$. We use VB program to develop this theory and for illustration of examples.

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Cited by 10 Pith papers

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    Computer programs are supplied to generate families of Type-2 isomorphic circulant graphs C_n(R) for m=2,3,5,7 and to demonstrate how Type-1 and Type-2 isomorphisms occur.

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    There are 960 triples of Type-2 isomorphic circulant graphs C_54(R) where each triple is Type-2 isomorphic with m=3.

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    Certain circulant graphs C_16(1,2,7) and C_16(2,3,5) are Type-2 isomorphic w.r.t. m=2, and for n>=2 families C_8n(R) with R={2,2s-1,4n-(2s-1)} and C_8n(S) are Type-2 isomorphic w.r.t. m=2 under stated conditions on n and s.

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    Two families of circulant graphs C_432(R) and C_6750(S) each possess Type-2 isomorphic variants for two values of m.

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    Proves that C_{np^3}(R^{np^3,x+yp}_i) for i=1 to p are Type-2 isomorphic w.r.t. m=p and form an Abelian group T2_{np^3,p} under the operation defined by index addition modulo p.

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