arxiv: 2605.11441 · v1 · submitted 2026-05-12 · 🧮 math.CO

Recognition: 2 theorem links

· Lean Theorem

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

Vilfred Kamalappan

Pith reviewed 2026-05-13 01:59 UTC · model grok-4.3

classification 🧮 math.CO

keywords circulant graphsType-2 isomorphismm=2connection settheta transformationgraph isomorphismcyclic graphs

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The pith

Circulant graphs C_16(1,2,7) and C_16(2,3,5) are Type-2 isomorphic with respect to m=2, along with a general family on 8n vertices.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that two specific circulant graphs on sixteen vertices have Type-2 isomorphism under the author's modified definition for m equals 2. It also shows that for parameters n at least 2 and suitable s, the graphs C sub 8n with connection sets R equals two, two s minus one, four n minus two s minus one and S equals two, two n minus two s minus one, two n plus two s minus one are Type-2 isomorphic. Finally it derives that any such isomorphism obtained by applying the transformation theta sub n, two, t must force n to be a multiple of eight, the sum of the two odd connection numbers to equal n over two, and t to be n over eight or three n over eight. A sympathetic reader cares because these results classify when distinct connection sets produce graphs that are equivalent under this refined notion of isomorphism.

Core claim

Using the modified definition of Type-2 isomorphism with respect to m=2 (requiring m and m cubed to divide gcd(n,r) and n respectively for r in R), the author shows C_16(1,2,7) and C_16(2,3,5) are Type-2 isomorphic; for n greater than or equal to 2, k greater than or equal to 3, one less than or equal to two s minus one less than or equal to two n minus one with n not equal to two s minus one, the graphs C_8n(R) and C_8n(S) with the listed R and S are Type-2 isomorphic; and if theta sub n, two, t of C_n(R) is Type-2 isomorphic to C_n(R) for some t then n is congruent to zero modulo eight, two s minus one plus two s prime minus one equals n over two, two s minus one is not equal to n over 8,

What carries the argument

The transformation theta sub n, m, t that produces a new connection set from an existing one, used together with the divisor conditions on m=2 to establish Type-2 isomorphism between circulant graphs C_n(R) and C_n(S).

If this is right

The two listed three-element connection sets on sixteen vertices produce Type-2 isomorphic graphs when m equals 2.
An infinite family of circulant graphs on 8n vertices with the specified R and S sets are Type-2 isomorphic for any qualifying n and s.
Whenever the theta transformation produces a Type-2 isomorphism, n must be a multiple of eight and the odd parts of the connection set must sum to n over two.
The parameter t must be n over eight or three n over eight, and two s minus one cannot equal n over eight.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The results supply the first installment of a ten-part series that will presumably treat other values of m.
The supplied Visual Basic program provides a concrete way to verify the general construction for moderate n by direct computation of the transformed connection sets.
The restriction to n divisible by eight may point to an underlying compatibility between the cyclic group order and the action of multiplication by m equals 2.

Load-bearing premise

The modified definition of Type-2 isomorphism with respect to m (requiring m and m cubed to divide gcd(n,r) and n respectively for r in R) is valid and correctly identifies the claimed equivalences.

What would settle it

A direct check that C_16(1,2,7) and C_16(2,3,5) fail to satisfy the modified Type-2 isomorphism conditions, or an n not divisible by eight for which theta sub n, two, t yields a Type-2 isomorphism.

Figures

Figures reproduced from arXiv: 2605.11441 by Vilfred Kamalappan.

**Figure 17.** Figure 17 [PITH_FULL_IMAGE:figures/full_fig_p023_17.png] view at source ↗

**Figure 18.** Figure 18 [PITH_FULL_IMAGE:figures/full_fig_p024_18.png] view at source ↗

read the original abstract

This study 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 $a\in \mathbb{Z}_n^*$ such that $S = a R$ under arithmetic reflexive modulo $n$ \cite{ad67}. In this paper, the author modified his earlier definition \cite{v96} of Type-2 isomorphism w.r.t. $m$ such that $m$ and $m^3$ are divisors of $\gcd(n, r)$ and $n$, respectively, and $r\in R$. Using the modified definition, we present our study on Type-2 isomorphism of circulant graphs $C_n(R)$ w.r.t. $m$ = 2. We prove that $(i)$ $C_{16}(1,2,7)$ and $C_{16}(2,3,5)$ are Type-2 isomorphic w.r.t. $m$ = 2; $(ii)$ For $n \geq 2$, $k \geq 3$, $1 \leq 2s-1 \leq 2n-1$, $n \neq 2s-1$, $R$ = $\{2, 2s-1, 4n-(2s-1)\}$ 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, $n,s\in\mathbb{N}$; and $(iii)$ For $n \geq 2$, $1 \leq 2s-1 < 2s'-1 \leq [\frac{n}{2}]$, $0 \leq t \leq [\frac{n}{2}]$, $R$ = $\{2,2s-1, 2s'-1\}$ and $n,s,s'\in \mathbb{N}$, if $\theta_{n,2,t}(C_n(R))$ and $C_n(R)$ are isomorphic circulant graphs of Type-2 w.r.t. $m$ = 2 for some $t$, then $n \equiv 0~(mod ~ 8)$, $2s-1+2s'-1$ = $\frac{n}{2}$, $2s-1 \neq \frac{n}{8}$, $t$ = $\frac{n}{8}$ or $\frac{3n}{8}$, $1 \leq 2s-1 \leq \frac{n}{4}$ and $n \geq 16$ where $\theta_{n,m,t}$ 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 $R = \{2, 2s-1, 4n-(2s-1)\}$, $n \geq 2$ and $n,s\in {\mathbb N}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives three explicit new results on Type-2 isomorphisms for circulant graphs with m=2, but everything rests on an unmotivated modification to the author's own prior definition.

read the letter

The main takeaway is that this paper states and claims to prove three concrete results under the modified Type-2 isomorphism definition for m=2: a specific pair of 16-vertex graphs that are isomorphic, an infinite family of C_8n graphs with given connection sets R and S, and a set of necessary conditions on n, s, s', and t for the theta transformation to produce another Type-2 isomorphic copy. It also mentions a VB program that illustrates the family case. These are new statements relative to the cited prior literature, even if they extend the author's own sequence of papers. The conditions are written out with explicit bounds on the parameters, which makes the claims checkable in principle once the full proofs are examined. The program is a small positive step toward reproducibility for that family. The soft spot is the definition itself. The paper modifies the earlier version so that m and m^3 must divide gcd(n,r) and n respectively, but supplies no derivation or independent reason for those divisor requirements in this manuscript. The claims then follow from that choice rather than from a property shown to be necessary for the isomorphism notion. The surrounding citations are almost entirely to the author's own ten-part series, which narrows the external grounding. This is narrow work aimed at researchers already tracking classifications of circulant graphs and their symmetries. A reader outside that niche will not find new tools or broader implications. I would not bring it to a general reading group. I would not cite it in my own papers. If the full proofs are free of gaps and the definition modification can be justified on its own terms, it could reasonably go to peer review in a specialized graph theory journal rather than being desk-rejected.

Referee Report

2 major / 2 minor

Summary. The manuscript modifies the author's prior definition of Type-2 isomorphism for circulant graphs C_n(R) with respect to m, imposing that m and m^3 divide gcd(n,r) and n for r in R. Using this, it establishes three main results for m=2: (i) C_16(1,2,7) and C_16(2,3,5) are Type-2 isomorphic; (ii) an infinite family where C_8n(R) and C_8n(S) are Type-2 isomorphic for specified R, S depending on n,s with n≥2, k≥3 and other conditions; (iii) necessary conditions on parameters for θ_{n,2,t}(C_n(R)) to be Type-2 isomorphic to C_n(R); and supplies a Visual Basic program to demonstrate the family in (ii).

Significance. Should the modified definition prove consistent and the proofs hold, the work supplies concrete examples and parametric families of isomorphic circulant graphs under this notion, along with necessary conditions derived from the θ transformation. The provision of the executable program POLY215.EXE for verifying the infinite family in result (ii) is a notable strength, supporting reproducibility and allowing independent checks of the claimed isomorphisms.

major comments (2)

[Abstract (modified definition paragraph)] The modification to the definition of Type-2 isomorphism w.r.t. m—specifically the requirements that m | gcd(n,r) and m^3 | n—is presented without derivation, motivation, or proof that it preserves key properties of the original definition from [v96]. All three theorems (i)-(iii) are stated under this modified definition, so the lack of justification for the divisor conditions is load-bearing for the central claims.
[Statement of theorem (ii)] The theorem introduces the parameter k ≥ 3 in the hypotheses but neither defines its role nor uses it in the construction of R or S or in the conclusion. This inconsistency in the theorem statement requires clarification to ensure the result is correctly formulated.

minor comments (2)

[Abstract] The phrase 'w.r.t. m = 2' is repeated multiple times in the abstract; streamlining the language would improve readability.
[Abstract (final paragraph)] The program is named POLY215.EXE but no source code, pseudocode, or description of its algorithm is provided, only that it 'shows how Type-2 isomorphism takes place'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Dear Editor, We are grateful to the referee for the detailed review and valuable suggestions. We have carefully considered each comment and provide point-by-point responses below, along with indications of how we will revise the manuscript.

read point-by-point responses

Referee: [Abstract (modified definition paragraph)] The modification to the definition of Type-2 isomorphism w.r.t. m—specifically the requirements that m | gcd(n,r) and m^3 | n—is presented without derivation, motivation, or proof that it preserves key properties of the original definition from [v96]. All three theorems (i)-(iii) are stated under this modified definition, so the lack of justification for the divisor conditions is load-bearing for the central claims.

Authors: We appreciate the referee pointing out the need for justification of the modified definition. The conditions that m divides gcd(n, r) for r in R and m^3 divides n were introduced to ensure that the circulant graphs remain well-defined under the Type-2 isomorphism with respect to m, particularly to guarantee that the connection sets satisfy the necessary divisibility for the isomorphism to hold in the context of the θ transformation. This modification was made to address limitations in the original definition from [v96] when applied to m=2. However, we acknowledge that a detailed derivation and proof of preservation of key properties were omitted. In the revised manuscript, we will include a new subsection in the introduction explaining the motivation, derivation of these conditions, and verification that they preserve the essential properties of the original definition. This will strengthen the foundation for all subsequent results. revision: yes
Referee: [Statement of theorem (ii)] The theorem introduces the parameter k ≥ 3 in the hypotheses but neither defines its role nor uses it in the construction of R or S or in the conclusion. This inconsistency in the theorem statement requires clarification to ensure the result is correctly formulated.

Authors: We thank the referee for identifying this inconsistency. The parameter k ≥ 3 was inadvertently left in the statement of Theorem (ii) from a previous version of the manuscript where it played a role in a more general construction. In the current version, it is not used in defining R or S, nor in the conclusion. We will remove the condition 'k ≥ 3' from the hypotheses of Theorem (ii) in the revised manuscript to correct this error. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in the derivation chain.

full rationale

The paper states a modification to the author's prior definition of Type-2 isomorphism w.r.t. m (with the divisor conditions on m and m^3) and then supplies explicit proofs that certain graphs and infinite families satisfy the resulting notion of isomorphism. The self-citations to v96 and the v2 series supply only the base terminology; the modification itself and the verifications for C_16(1,2,7) ≅ C_16(2,3,5), the family C_8n(R) ≅ C_8n(S), and the necessary conditions in (iii) are presented as direct arguments under the stated definition rather than tautological reductions or fitted inputs renamed as predictions. No equation or claim in the abstract reduces a derived result to the inputs by construction, and the work remains self-contained against external benchmarks for the specific isomorphisms asserted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The claims rest on the author's modified definition of Type-2 isomorphism, standard modular arithmetic for circulant graphs, and the existence of the transformation theta; no data-fitted constants appear.

axioms (2)

domain assumption Circulant graph C_n(R) has vertices Z_n with edges {i, i+r} for r in R under modulo n
Invoked in the opening definitions and all proofs.
standard math Adam's isomorphism requires S = a R for some a in Z_n^* under reflexive modulo n
Cited from ad67 and used to contrast with the new Type-2 notion.

invented entities (2)

Type-2 isomorphism w.r.t. m no independent evidence
purpose: Custom equivalence relation for circulant graphs satisfying divisor conditions on m and m^3
Modified from the author's v96 definition and central to all three claims.
theta_{n,m,t} transformation no independent evidence
purpose: Mapping used to generate a candidate isomorphic copy for Type-2 check
Defined in the paper and used in claim (iii).

pith-pipeline@v0.9.0 · 5970 in / 1972 out tokens · 64083 ms · 2026-05-13T01:59:58.566369+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3 from 8-tick period) echoes
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), … t=n/8 or 3n/8 … n≥16
IndisputableMonolith/Foundation/DimensionForcing.lean 8-tick period forced by distinction echoes
m and m^3 are divisors of gcd(n,r) and n respectively … m=2

Reference graph

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