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arxiv: 2508.09384 · v2 · submitted 2025-08-12 · 🧮 math.CO

A study on Type-2 isomorphic circulant graphs. Part 2: Type-2 isomorphic circulant graphs of orders 16, 24, 27

Vilfred Kamalappan This is my paper

Pith reviewed 2026-05-18 22:23 UTC · model grok-4.3

classification 🧮 math.CO
keywords circulant graphsType-2 isomorphismgraph isomorphismorder 16order 24order 27
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The pith

Modified definition of Type-2 isomorphism identifies 8 pairs of circulant graphs on 16 vertices, 32 pairs on 24 vertices, and 12 triples on 27 vertices.

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

This paper applies a further modified definition of Type-2 isomorphism for circulant graphs C_n(R) with respect to a parameter m. The modification adds the requirements that m > 1 divides gcd(n, r), m cubed divides n, and r belongs to the connection set R. With this change the calculations become simpler, and the paper enumerates the resulting isomorphic pairs and triples for the three concrete orders 16, 24 and 27. The explicit counts obtained are 8 pairs for order 16, 32 pairs for order 24, and 12 triples for order 27. A reader interested in systematic classification of circulant graphs would care because the numbers give verifiable data points that can be checked by direct computation on small vertex sets.

Core claim

Using the modified definition of Type-2 isomorphism of circulant graphs C_n(R) w.r.t. m, where m > 1 divides gcd(n, r), m^3 divides n and r in R, the paper obtains Type-2 isomorphic circulant graphs of orders 16, 24 and 27 and shows that the total number of pairs of Type-2 isomorphic circulant graphs of orders 16 and 24 are 8 and 32, respectively and the total number of triples of Type-2 isomorphic circulant graphs of order 27 are 12.

What carries the argument

The further modified definition of Type-2 isomorphism w.r.t. m that requires m > 1 divides gcd(n, r), m^3 divides n, and r in R, which simplifies the identification of isomorphic pairs and triples.

If this is right

  • Exactly 8 pairs of Type-2 isomorphic circulant graphs exist for order 16.
  • Exactly 32 pairs of Type-2 isomorphic circulant graphs exist for order 24.
  • Exactly 12 triples of Type-2 isomorphic circulant graphs exist for order 27.
  • The modified conditions reduce the computational work needed to list the isomorphic copies for these three orders.

Where Pith is reading between the lines

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

  • The same style of modification may make enumeration feasible for additional small orders that satisfy the divisibility pattern.
  • The reported counts supply concrete benchmarks that independent computational isomorphism software could confirm or adjust.
  • The approach of tailoring an isomorphism definition to exploit number-theoretic constraints on n and r could be tested on other families of vertex-transitive graphs.

Load-bearing premise

The added divisibility conditions on m correctly capture exactly the Type-2 isomorphisms that exist for the chosen orders without omitting any or adding extraneous ones.

What would settle it

An exhaustive check of all possible connection sets for n equal to 16, 24 and 27, followed by direct verification of graph isomorphism, that returns a different total number of pairs or triples than 8, 32 and 12 respectively.

read the original abstract

This study is the $2^{nd}$ part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10}. Definition of Type-2 isomorphism of circulant graphs $C_n(R)$ w.r.t. $m$ was further modified by the author by considering $m > 1$ divides $\gcd(n, r)$, $m^3$ divides $n$ and $r\in R$ and studied Type-2 isomorphic circulant graphs w.r.t. $m$ = 2. This modification simplifies our calculations while finding isomorphic circulant graphs of Type-2. In this paper, using the modified definition \ref{d4.2}, we obtain Type-2 isomorphic circulant graphs of orders 16, 24 and 27 and show that the total number of pairs of Type-2 isomorphic circulant graphs of orders 16 and 24 are 8 and 32, respectively and the total number of triples of Type-2 isomorphic circulant graphs of order 27 are 12.

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.

Referee Report

2 major / 1 minor

Summary. This paper is the second installment in a ten-part series on Type-2 isomorphic circulant graphs. It further modifies the definition of Type-2 isomorphism for circulant graphs C_n(R) by imposing the conditions that m > 1 divides gcd(n, r), m^3 divides n, and r ∈ R (with focus on m = 2). Using this modified Definition 4.2, the author enumerates Type-2 isomorphic families and reports exactly 8 pairs for order 16, 32 pairs for order 24, and 12 triples for order 27.

Significance. If the modified definition can be shown to preserve the original Type-2 isomorphism classes, the reported counts would supply concrete, order-specific data on the frequency of such isomorphic circulant graphs for small n. This could serve as a reference point for future classification efforts in circulant graph theory, though the manuscript provides no independent verification or machine-checked enumeration to strengthen the result.

major comments (2)
  1. [Definition 4.2] Definition 4.2: The additional constraints (m > 1 divides gcd(n, r), m^3 divides n, and r ∈ R) are presented as a simplification of the prior Type-2 notion, yet no lemma or argument establishes that this restriction is equivalent—i.e., that it neither excludes valid Type-2 isomorphisms nor introduces extraneous pairs/triples. Because the reported totals (8, 32, 12) are obtained directly from this definition, the equivalence is load-bearing for the central claims.
  2. [Results sections for n=16, 24, 27] Enumeration results (orders 16, 24, 27): The manuscript asserts the final counts without supplying the explicit list of graphs, the enumeration algorithm, or any verification steps. For these small orders an omitted case or overcount would directly change the reported numbers, and the absence of such details prevents independent checking of exhaustiveness.
minor comments (1)
  1. The paper refers to a ten-part series but supplies only minimal cross-references to how the current modification interacts with definitions established in the earlier parts.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We respond to each major comment below and indicate the changes we will make in the revised version.

read point-by-point responses

  1. Referee: [Definition 4.2] Definition 4.2: The additional constraints (m > 1 divides gcd(n, r), m^3 divides n, and r ∈ R) are presented as a simplification of the prior Type-2 notion, yet no lemma or argument establishes that this restriction is equivalent—i.e., that it neither excludes valid Type-2 isomorphisms nor introduces extraneous pairs/triples. Because the reported totals (8, 32, 12) are obtained directly from this definition, the equivalence is load-bearing for the central claims.

    Authors: We appreciate the referee's observation. In this part of the series, Definition 4.2 introduces a further restricted version of Type-2 isomorphism specifically to make explicit enumeration feasible for the small orders 16, 24, and 27. The manuscript does not claim or prove equivalence to the definition used in earlier parts; the restriction is presented as a computational simplification. The reported counts (8 pairs, 32 pairs, 12 triples) are obtained strictly under the conditions of this modified definition. In the revision we will add an explicit statement in the introduction and immediately after Definition 4.2 clarifying that this is a working definition adopted for the present study and that relating the restricted counts back to the original Type-2 notion is reserved for later papers in the series. revision: yes

  2. Referee: [Results sections for n=16, 24, 27] Enumeration results (orders 16, 24, 27): The manuscript asserts the final counts without supplying the explicit list of graphs, the enumeration algorithm, or any verification steps. For these small orders an omitted case or overcount would directly change the reported numbers, and the absence of such details prevents independent checking of exhaustiveness.

    Authors: We agree that greater transparency is needed for these small orders. Although the manuscript describes the overall approach, it does not include the explicit lists of pairs/triples or a detailed enumeration procedure. In the revised version we will add an appendix containing the complete lists of the 8 pairs for order 16, the 32 pairs for order 24, and the 12 triples for order 27. We will also include a concise description of the enumeration algorithm: systematic generation of admissible connection sets R that satisfy the divisibility conditions of Definition 4.2, followed by direct verification of the isomorphism relation. This will enable independent checking. We will additionally note the manual verification steps used to confirm exhaustiveness for these orders. revision: yes

Circularity Check

0 steps flagged

No circularity: counts are direct enumerations under the paper's stated definition

full rationale

The manuscript cites its own prior parts [v2-1]–[v2-10] only to establish the modified Definition 4.2 (m > 1 dividing gcd(n,r), m³ dividing n, r ∈ R, studied for m=2). It then applies this definition to list and count the qualifying circulant graphs on the finite vertex sets of orders 16, 24 and 27. The reported totals (8 pairs, 32 pairs, 12 triples) are therefore the direct output of exhaustive classification under the chosen criteria; they do not reduce to the definition by algebraic identity, statistical fitting, or self-referential uniqueness theorem. No prediction, parameter estimation, or load-bearing lemma is present whose validity collapses back onto the inputs. The work is self-contained as an enumeration exercise once the definition is fixed.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, invented entities, or non-standard axioms are stated. The work relies on the author's prior modification of the Type-2 isomorphism definition and standard circulant-graph constructions.

axioms (2)
  • standard math Circulant graphs C_n(R) are defined in the usual way with vertices on a cycle and edges determined by the set R.
    Invoked throughout the abstract and the series.
  • domain assumption The further modified definition of Type-2 isomorphism w.r.t. m is the correct and useful notion for this study.
    Stated as the basis for all calculations in the paper.

pith-pipeline@v0.9.0 · 5735 in / 1617 out tokens · 52114 ms · 2026-05-18T22:23:56.570734+00:00 · methodology

discussion (0)

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Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A study on Type-2 isomorphic circulant graphs. Part 3: 384 pairs of Type-2 isomorphic circulant graphs $C_{32}(R)$

    math.CO 2026-05 unverdicted novelty 5.0

    All 384 pairs of Type-2 isomorphic circulant graphs C_32(R) have been obtained.

  2. A study on Type-2 isomorphic circulant graphs. Part 6: Abelian groups $(T2_{n,m}(C_n(R)), \circ)$ and $(V_{n,m}(C_n(R)), \circ)$

    math.CO 2026-05 unverdicted novelty 4.0

    V_{n,m}(C_n(R)) forms an Abelian group under ∘ and T2_{n,m}(C_n(R)) is a subgroup, where T2 collects C_n(R) and all its Type-2 isomorphic copies w.r.t. m.

  3. A study on Type-2 isomorphic circulant graphs. Part 4: 960 triples of Type-2 isomorphic circulant graphs $C_{54}(R)$

    math.CO 2026-05 unverdicted novelty 4.0

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

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

    math.CO 2026-05 unverdicted novelty 4.0

    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.

  5. A Study on Type-2 Isomorphic Circulant Graphs: Part 8: $C_{432}(R)$, $C_{6750}(S)$ -- each has 2 types of Type-2 isomorphic circulant graphs

    math.CO 2026-05 unverdicted novelty 3.0

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

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

    math.CO 2026-05 unverdicted novelty 2.0

    Enumeration yields 18 Type-2 isomorphic pairs for n=48, 72 pairs for n=96, and 27 triples for n=81 among circulant graphs C_n with 3 or 4 generators.

Reference graph

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