A study on Type-2 isomorphic circulant graphs. Part 1: Type-2 isomorphic circulant graphs C_n(R) w.r.t. m = 2

@misc{kamalappan2026study,
  title = {{A study on Type-2 isomorphic circulant graphs. Part 1: Type-2 isomorphic circulant graphs $C_n(R)$ w.r.t. $m$ = 2}},
  author = {Vilfred Kamalappan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.11441},
  eprint = {2605.11441},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.11441},
  note = {CC BY-NC-SA 4.0},
}

TY  - UNPB
TI  - A study on Type-2 isomorphic circulant graphs. Part 1: Type-2 isomorphic circulant graphs C_n(R) w.r.t. m = 2
AU  - Vilfred Kamalappan
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2605.11441
UR  - https://arxiv.org/abs/2605.11441
AB  - This article is the first part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10}. Circulant graphs C_n(R) and C_n(S) are said to be \emph{Adam's isomorphic} if there exist some a∈ ℤ_n^* such that S = a R under arithmetic reflexive modulo n \cite{ad67}. 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). The author \cite{v96} further modify the definition to m and m^3 are divisors of gcd(n, r) and n, respectively, and r∈ R. Using the modified definition \ref{d16}, we present our study on Type-2 isomorphism of circulant graphs C_n(R) w.r.t. m = 2. We reprove that (i) C_16(1,2,7) and C_16(2,3,5) are Type-2 isomorphic w.r.t. m = 2; (ii) For n ≥ 2, k ≥ 3, 1 ≤ 2s−1 ≤ 2n−1, n ≠ 2s−1, R = {2, 2s−1, 4n−(2s−1)} and S = {2, 2n−(2s−1), 2n+2s−1}, C_8n(R) and C_8n(S) are Type-2 isomorphic w.r.t. m = 2, n,s∈ℕ; and (iii) For n ≥ 2, 1 ≤ 2s−1 < 2s'−1 ≤ [n/2], 0 ≤ t ≤ [n/2], R = {2,2s−1, 2s'−1} and n,s,s'∈ ℕ, if θ_n,2,t(C_n(R)) and C_n(R) are isomorphic circulant graphs of Type-2 w.r.t. m = 2 for some t, then n ≡ 0 (mod 8), 2s−1+2s'−1 = n/2, 2s−1 ≠ n/8, t = n/8 or 3n/8, 1 ≤ 2s−1 ≤ n/4 and n ≥ 16 where θ_n,m,t is a transformation used to define Type-2 isomorphism of a circulant graph. At the end, we present a VB program POLY215.EXE which shows how Type-2 isomorphism w.r.t. m = 2 of C_8n(R) takes place for R = {2, 2s−1, 4n−(2s−1)}, n ≥ 2 and n,s∈ ℕ.
ER  -