arxiv: 2605.12390 · v1 · submitted 2026-05-12 · 🧮 math.CO

Recognition: no theorem link

A study on Type-2 isomorphic circulant graphs. Part 5: Type-2 isomorphic circulant graphs of orders 48, 81, 96

Vilfred Kamalappan

Pith reviewed 2026-05-13 03:42 UTC · model grok-4.3

classification 🧮 math.CO

keywords circulant graphsType-2 isomorphismgraph isomorphismenumerationorders 48 81 96

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The pith

There are exactly 18 pairs of Type-2 isomorphic circulant graphs on 48 vertices, 72 such pairs on 96 vertices, and 27 triples on 81 vertices.

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

This paper extends an ongoing enumeration of circulant graphs that are isomorphic under a specific relation called Type-2 isomorphism. It examines graphs of the form C_n with three or four connection differences for the concrete orders 48, 81, and 96. The authors apply the counting procedure developed in prior parts of the series and report precise totals for pairs and triples that satisfy the isomorphism condition. These numbers complete the classification for the selected orders and supply concrete data on how connection sets can be transformed while preserving the graph. The results allow direct comparison of isomorphism counts across different vertex orders.

Core claim

The total number of pairs of isomorphic circulant graphs of Type-2 with respect to m=2 of the forms C_48(r1,r2,r3) and C_48(s1,s2,s3) is 18, the corresponding total for n=96 is 72, and the total number of triples of isomorphic circulant graphs of Type-2 with respect to m=3 of the form C_81(x1,x2,x3), C_81(y1,y2,y3) and C_81(z1,z2,z3) is 27.

What carries the argument

Type-2 isomorphism with respect to an integer parameter m, which identifies when distinct connection sets produce identical circulant graphs after a specific scaling or modular transformation.

Load-bearing premise

The enumeration procedure and definition of Type-2 isomorphism developed in the earlier parts of the series correctly identify all isomorphic instances for these three orders without omissions or false positives.

What would settle it

Discovery of one additional pair of Type-2 isomorphic graphs on 48 vertices that satisfies the definition but is absent from the reported 18, or failure to locate any of the claimed pairs under independent verification, would falsify the counts.

read the original abstract

This study is the $5^{th}$ 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 4. Here, we study Type-2 isomorphic circulant graphs of $C_{48}(r_1,r_2,r_3)$, $C_{81}(r_1,r_2,r_3)$ and $C_{96}(r_1,r_2,r_3,r_4)$. We find that the total number of pairs of isomorphic circulant graphs of Type-2 w.r.t. $m$ = 2 of the forms $C_{n}(r_1,r_2,r_3)$ and $C_{n}(s_1,s_2,s_3)$ are 18 and 72 for $n$ = 48, 96, respectively and the total number of triples of isomorphic circulant graphs of Type-2 w.r.t. $m$ = 3 of the form $C_{81}(x_1,x_2,x_3)$, $C_{81}(y_1,y_2,y_3)$ and $C_{81}(z_1,z_2,z_3)$ are 27.

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 paper adds three specific counts (18 pairs for n=48, 72 for n=96, 27 triples for n=81) to an ongoing enumeration series but supplies no new methods, samples, or independent checks on the results.

read the letter

The main thing here is that the paper reports concrete totals for Type-2 isomorphic circulant graphs on three fixed orders: 18 pairs for n=48, 72 pairs for n=96, and 27 triples for n=81. These numbers are presented as the output of applying the definition and procedure from the earlier parts of the series to these cases. If the prior work is correct, the counts are new data points for those specific n values and connection-set sizes. The paper does the straightforward job of extending the enumeration to these orders without claiming general theorems or new techniques. That consistency is the main positive. The central limitation is the complete absence of supporting detail in this installment. The abstract states the totals but includes no sample connection sets, no listing of the actual isomorphic instances, no description of how the enumeration was carried out for these orders, and no argument that the procedure captured every case without false positives. All of the weight rests on the unexamined correctness of parts 1-4. Without those details or any cross-check, the numbers cannot be assessed directly. This work is only useful to readers who are already following the full series on Type-2 circulant graph isomorphisms. Anyone outside that narrow track will find little to engage with. I would not bring it to a reading group. I would not cite it unless I needed those exact counts for some follow-on calculation. It could be sent for peer review if a journal has already committed to the series, but the lack of verification steps means it would likely require substantial additions to stand alone.

Referee Report

1 major / 0 minor

Summary. This paper is the fifth part of a ten-part series on Type-2 isomorphic circulant graphs. It enumerates Type-2 isomorphisms among 3-regular circulant graphs C_n(r1,r2,r3) for n=48 and n=96 (reporting 18 and 72 isomorphic pairs, respectively) and among 3-regular circulant graphs on 81 vertices (reporting 27 isomorphic triples). The results are presented as direct counts obtained by applying the Type-2 isomorphism definition and enumeration procedure developed in earlier parts of the series.

Significance. If the counts are accurate and complete, the work supplies concrete numerical data on the occurrence of Type-2 isomorphisms for three specific orders. These data points can serve as benchmarks for testing general conjectures about the structure of automorphism groups or isomorphism classes of circulant graphs and may help identify patterns across the series.

major comments (1)

Abstract: the central claims are the numerical totals (18 pairs for n=48, 72 pairs for n=96, 27 triples for n=81), yet the manuscript supplies neither the explicit lists of connection sets forming these pairs/triples, nor any sample isomorphisms, nor a self-contained description of the enumeration steps or completeness check performed for these orders. This omission is load-bearing for the central claim because the reported counts cannot be verified or reproduced from the text alone.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review and for highlighting the need for greater verifiability of the reported counts. We address the major comment below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

Referee: Abstract: the central claims are the numerical totals (18 pairs for n=48, 72 pairs for n=96, 27 triples for n=81), yet the manuscript supplies neither the explicit lists of connection sets forming these pairs/triples, nor any sample isomorphisms, nor a self-contained description of the enumeration steps or completeness check performed for these orders. This omission is load-bearing for the central claim because the reported counts cannot be verified or reproduced from the text alone.

Authors: We acknowledge that the current presentation states the counts without supplying the full connection-set lists, representative isomorphisms, or a self-contained recap of the enumeration procedure. The underlying algorithm and definition of Type-2 isomorphism are developed in Parts 1–4 of the series, and the present work applies that framework to the three new orders. To address the referee’s concern, we will revise the manuscript by adding a dedicated subsection that (i) briefly recapitulates the enumeration steps and completeness argument tailored to n = 48, 81, and 96, (ii) provides at least one explicit sample isomorphism for each reported count, and (iii) indicates that the complete lists of connection sets are available as supplementary material or upon request. Because the number of pairs for n = 96 is 72, embedding every list in the main text would be impractical, but the added samples and procedural outline will allow readers to reproduce the method and verify the approach. revision: yes

Circularity Check

0 steps flagged

No circularity: direct enumeration results from prior procedure

full rationale

The paper reports counts of isomorphic pairs (18 for n=48, 72 for n=96) and triples (27 for n=81) as outcomes of applying the Type-2 isomorphism definition and enumeration procedure developed in the cited prior parts of the series. No equations, derivations, predictions, or first-principles steps appear in the provided text that reduce by construction to inputs, self-definitions, or fitted parameters. The central claims are presented as enumeration findings rather than quantities forced by the paper's own definitions or self-citations. Self-citation to the series exists but is not load-bearing in a circular sense here, as the results are new applications that remain externally falsifiable via independent computation of the connection sets for these fixed orders.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard definition of circulant graphs and the Type-2 isomorphism notion introduced in the earlier parts of the series; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

standard math Standard definition and basic properties of circulant graphs C_n(S) and graph isomorphism
Invoked throughout the abstract and the series it continues.

pith-pipeline@v0.9.0 · 5549 in / 1283 out tokens · 113836 ms · 2026-05-13T03:42:36.693524+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 1 internal anchor

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