arxiv: 2605.11466 · v1 · submitted 2026-05-12 · 🧮 math.CO

Recognition: 1 theorem link

· Lean Theorem

A study on Type-2 isomorphic circulant graphs. Part 3: 384 pairs of Type-2 isomorphic circulant graphs C₃₂(R)

Vilfred Kamalappan

Pith reviewed 2026-05-13 01:43 UTC · model grok-4.3

classification 🧮 math.CO

keywords circulant graphsType-2 isomorphismisomorphic graphsorder 32connection setsgraph enumerationcyclic graphs

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The pith

This paper identifies all 384 pairs of Type-2 isomorphic circulant graphs of order 32.

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

The work continues a series on Type-2 isomorphisms among circulant graphs, which are graphs whose edges are determined by a fixed set of steps around a cycle. It exhaustively determines every pair of distinct connection sets R and S such that the graphs C_32(R) and C_32(S) are isomorphic under the Type-2 definition. A reader would care because this supplies a complete, order-specific catalogue of when two such highly symmetric graphs coincide despite different generators. The result makes it possible to check general statements about circulant graph isomorphisms against a concrete, finite case of size 32.

Core claim

The authors obtain and present the complete collection of 384 pairs of connection sets R and S for which the circulant graphs C_32(R) and C_32(S) are Type-2 isomorphic, where each pair consists of distinct subsets of {1,2,...,15} that satisfy the required conditions for this form of isomorphism.

What carries the argument

The Type-2 isomorphism condition applied to the connection sets R of the circulant graphs C_n(R), which equates two graphs when their generating steps can be matched under a specific non-standard mapping that preserves the cyclic ordering.

Load-bearing premise

The enumeration procedure checks every possible connection set against every other and correctly identifies precisely those pairs that meet the Type-2 isomorphism condition, with no omissions or inclusions of invalid cases.

What would settle it

An exhaustive independent search over all possible connection sets for order 32 that finds either a Type-2 isomorphic pair absent from the reported list or a listed pair that fails the isomorphism condition.

read the original abstract

This study is the $3^{rd}$ part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10} and is a continuation of Part 2. Here, we obtain all the 384 pairs of Type-2 isomorphic circulant graphs of order 32.

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

This part of the series states there are exactly 384 pairs for order 32 but supplies no method, code, or sample data to support the enumeration.

read the letter

The main thing here is that the paper reports a count of 384 pairs of Type-2 isomorphic circulant graphs C_32(R). It continues the author's multi-part study by giving this specific number for n=32, which does not appear in the cited prior parts. That concrete enumeration result is what the work adds. The author has been working through these cases in sequence, and this installment supplies the next data point in a narrow corner of circulant graph isomorphism. The computation itself is feasible, since the number of valid connection sets is only around 32k. The paper does at least focus on a fixed small order where an exhaustive approach could in principle be carried out. The central problem is that none of the supporting details are present. The text gives no description of the enumeration algorithm, no pseudocode, no list or checksum of the pairs found, and no argument showing that every possible R was checked without omissions or misclassifications. The definition of Type-2 isomorphism is presumably taken from earlier installments, but even so the application to all cases needs to be shown. Without that, the claimed total cannot be checked against the paper's own content. The citation pattern stays inside the series and does not engage much with outside literature on circulant graphs. This work is only relevant to the small group of researchers already following the ten-part study on Type-2 isomorphisms. A general combinatorialist or graph theorist would get nothing usable from it. I would not bring the paper to a reading group. I would not cite the 384 figure in my own work. It does not deserve peer review in its current form because the result is presented without the evidence needed to evaluate it.

Referee Report

1 major / 0 minor

Summary. The paper, the third installment in a ten-part series on Type-2 isomorphic circulant graphs, asserts that it has enumerated and obtained all 384 pairs of Type-2 isomorphic circulant graphs C_{32}(R) of order 32.

Significance. A verified complete enumeration for n=32 would supply a concrete, finite dataset that could support pattern detection or inductive conjectures for Type-2 isomorphisms in circulant graphs of larger even order, building on the preceding parts of the series.

major comments (1)

The central claim of exactly 384 pairs depends on exhaustive generation of all valid connection sets R (subsets of {1,...,16} satisfying R=-R) together with correct classification under the Type-2 isomorphism definition. The manuscript supplies neither a description of the enumeration algorithm, pseudocode, verification steps, nor any checksums, sample output, or completeness argument, rendering the count unverifiable from the text alone.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review and for identifying an important point regarding the verifiability of our results. We address the major comment below and will revise the manuscript to incorporate additional details on the enumeration process.

read point-by-point responses

Referee: The central claim of exactly 384 pairs depends on exhaustive generation of all valid connection sets R (subsets of {1,...,16} satisfying R=-R) together with correct classification under the Type-2 isomorphism definition. The manuscript supplies neither a description of the enumeration algorithm, pseudocode, verification steps, nor any checksums, sample output, or completeness argument, rendering the count unverifiable from the text alone.

Authors: We agree that the manuscript as submitted does not contain a self-contained description of the enumeration algorithm or associated verification materials. The general method for generating valid connection sets R and classifying Type-2 isomorphisms was introduced in Part 1 of the series, with this part focusing on the specific results for order 32. To address the referee's concern and improve verifiability, we will add a dedicated subsection to the revised manuscript that outlines the algorithm, provides pseudocode for the exhaustive generation and classification steps, describes verification procedures (including consistency checks against smaller-order cases from prior parts), and supplies checksums together with sample outputs. This addition will furnish the requested completeness argument without changing the reported count of 384 pairs. revision: yes

Circularity Check

0 steps flagged

Enumeration result with no self-referential derivation or fitted prediction

full rationale

The paper states that it obtains all 384 pairs of Type-2 isomorphic circulant graphs C_32(R) as the outcome of an enumeration procedure over connection sets. No equations, definitions, or derivations are presented that reduce the claimed count to a fitted parameter, a self-citation chain, or an input quantity by construction. Prior parts in the series are cited only to establish the definition of Type-2 isomorphism and the overall study framework; the specific numerical result remains an independent computational output that does not loop back to itself. This matches the default case of a non-circular enumeration claim.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard definitions of circulant graphs and the Type-2 isomorphism relation introduced in earlier parts of the series. No new free parameters, ad-hoc axioms, or invented entities are introduced in this installment.

axioms (1)

standard math Standard group-theoretic definition of circulant graphs C_n(R) and the Type-2 isomorphism relation.
Invoked throughout the series; the present paper treats these as given background.

pith-pipeline@v0.9.0 · 5353 in / 1072 out tokens · 49342 ms · 2026-05-13T01:43:00.012401+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/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
we obtain all the 384 pairs of Type-2 isomorphic circulant graphs of order 32

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

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V. Vilfred Kamalappan and P. Wilson,A study on Type-2 Isomorphic Circulant Graphs. Part 10: Type-2 isomorphicC np3 (R)w.r.t.m=pand related groups. Preprint. 20 pages Department of Mathematics, Central University of Kerala, Periye, Kasaragod, Kerala, India - 671 316. Email address:vilfredkamal@gmail.com

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