arxiv: 2605.14140 · v1 · pith:HCG6RWYZnew · submitted 2026-05-13 · 🧮 math.CO

A study on Type-2 isomorphic circulant graphs. PART 9: Computer programs to show Type-1 \& -2 isomorphic circulant graphs

Vilfred Kamalappan , Wilson Peraprakash This is my paper

Pith reviewed 2026-05-15 02:11 UTC · model grok-4.3

classification 🧮 math.CO

keywords circulant graphsType-2 isomorphismgraph isomorphismcomputer programsC++Visual Basic

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The pith

Computer programs generate families of Type-2 isomorphic circulant graphs for m values 2, 3, 5, 7 and odd primes p.

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

This paper describes C++ and Visual Basic programs developed to locate and illustrate Type-2 isomorphic circulant graphs C_n(R). These graphs become isomorphic under a specific multiplier m when m shares a divisor greater than 1 with n. The programs produce explicit families for m equal to 2, 3, 5, or 7 across all natural numbers n, and additional families for m equal to an odd prime p when n is a multiple of p cubed. A reader would care because finding such isomorphisms by hand is laborious, and the tools supply concrete examples that support further theoretical work on these graphs.

Core claim

The authors state that their C++ program has produced families of Type-2 isomorphic C_n(R) with respect to m=2,3,5,7 for every natural number n, together with families of C_{n p^3}(R) with respect to m=p for every odd prime p. They also supply the Visual Basic program POLY415.EXE that displays how Type-1 and Type-2 isomorphisms operate on a given circulant graph and that checks or finds such graphs for any chosen order.

What carries the argument

The C++ search program that enumerates and verifies Type-2 isomorphisms of circulant graphs C_n(R) with respect to a chosen multiplier m.

Load-bearing premise

The programs correctly implement the definitions of Type-1 and Type-2 isomorphism from the cited earlier papers without coding errors or missed cases.

What would settle it

Direct comparison of the program's output for small explicit n against hand-verified isomorphisms; any mismatch for a known pair would falsify the reliability of the generated families.

Figures

Figures reproduced from arXiv: 2605.14140 by Vilfred Kamalappan, Wilson Peraprakash.

**Figure 10.** Figure 10: It indicates C27(4, 5, 12, 13) = C27(4(1, 3, 8, 10)) is Type-1 isomorphic to C27(1, 3, 8, 10). Repeat this step to get more such graphs, if exist. (vi) Click Exist when you want to come out of the trail. (vii) Repeat the above steps for another circulant graph. And so on. 5. Conclusion In general, computational complexity on checking isomorphism of graphs increases with order and degree of the graphs when… view at source ↗

read the original abstract

Elspas and Turner \cite{eltu} raised a question on the isomorphism of $C_{16}(1,3,7)$ and $C_{16}(2,3,5)$ and Vilfred \cite{v96} gave its answer by defining Type-2 isomorphism of $C_n(R)$ w.r.t. $m$ $\ni$ $m$ = $\gcd(n, r) > 1$, $r\in R$ and $r,n\in\mathbb{N}$ and studied such graphs for $m$ = 2 in \cite{v13,v20}. But obtaining Type-2 isomorphic circulant graphs is not easy. Using a $C^{++}$ computer program, the authors obtained families of Type-2 isomorphic $C_{n}(R)$ w.r.t. $m$ = 2,3,5,7 for $n\in\mathbb{N}$ as well as $C_{np^3}(R)$ w.r.t. $m$ = $p$ for $n\in\mathbb{N}$ and $p$ is an odd prime. In this paper, we present the $C^{++}$ program and also a VB program POLY415.EXE which is used to show how Type-1 and Type-2 isomorphisms of a circulant graph take place as well as for checking and finding Type-1 and Type-2 circulant graphs of a given order and is very useful to develop its theory on Type-2 isomorphic circulant graphs \cite{v2-1}-\cite{v2-10}.

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 mostly releases C++ search code for Type-2 isomorphic circulants on specific m values, but gives almost no verification or sample outputs to show the code actually works.

read the letter

The core offering is a pair of programs: a C++ searcher that claims to locate families of Type-2 isomorphic C_n(R) for m=2,3,5,7 and for C_{n p^3}(R) when m=p odd prime, plus a Visual Basic helper for checking Type-1 versus Type-2 cases. That is the new piece relative to the earlier papers in the series; the definitions themselves are carried over from prior work. The code is printed, which at least lets readers inspect the logic directly. For someone already working on circulant isomorphism questions, having ready-made search tools could save time on brute-force enumeration. The programs rest on explicit definitions rather than fitted parameters, so there is no obvious circularity inside this manuscript. The main weakness is the absence of any concrete output. The abstract says families were found, yet the text supplies neither listed examples nor a run on the well-known Elspas-Turner case C_16(1,3,7) ≅ C_16(2,3,5). Without those checks or even a short execution trace, it is impossible to judge whether the gcd and connection-set handling is correct or whether off-by-one errors are producing phantom families. That gap makes the central claim rest entirely on trust in the implementation. This work is aimed at the small group already following the Type-2 circulant series. A reader outside that niche will not get much from it. It is coherent on its own terms and shows straightforward computational thinking, so it clears the bar for serious refereeing even though the evidence is thin. I would not cite it myself unless the authors later publish the actual families with verification. A journal could reasonably send it out for review, but only with a clear request that the authors add test cases and sample results before acceptance.

Referee Report

3 major / 2 minor

Summary. The manuscript presents a C++ program and a Visual Basic helper program (POLY415.EXE) that are claimed to detect and illustrate Type-1 and Type-2 isomorphisms of circulant graphs C_n(R), and states that these programs have produced families of Type-2 isomorphic graphs for m=2,3,5,7 and for C_{n p^3}(R) with m=p an odd prime.

Significance. If the programs faithfully implement the Type-1/Type-2 definitions from the cited prior papers in the series, the work supplies concrete computational tools that could accelerate the enumeration of isomorphic families and thereby support further theoretical development of Type-2 isomorphism for circulant graphs.

major comments (3)

Abstract: the claim that 'families were obtained via the programs' is unsupported because the manuscript supplies neither any explicit listing of the families found for m=2,3,5,7 nor for the p-odd-prime case, nor any sample output.
C++ program presentation: no line-by-line correspondence is given between the code and the formal definitions of Type-1 and Type-2 isomorphism from the referenced earlier papers; in particular, the gcd and connection-set mapping logic is not shown to match the definitions.
Verification section (or lack thereof): the manuscript contains no execution trace on the canonical Elspas-Turner example C_16(1,3,7) ≅ C_16(2,3,5), leaving open the possibility of off-by-one or gcd-handling errors that would silently produce spurious families.

minor comments (2)

Abstract: the notation 'C^{++}' should be written as 'C++'.
References: the citations v2-1 through v2-10 should be expanded with full bibliographic details and DOIs or arXiv identifiers where available.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address each major comment below and will incorporate the requested clarifications and additions in a revised version of the manuscript.

read point-by-point responses

Referee: Abstract: the claim that 'families were obtained via the programs' is unsupported because the manuscript supplies neither any explicit listing of the families found for m=2,3,5,7 nor for the p-odd-prime case, nor any sample output.

Authors: We agree that the abstract claim would be strengthened by concrete evidence. In the revision we will add representative sample outputs from the C++ program (and the POLY415.EXE helper) that explicitly list families of Type-2 isomorphic circulant graphs for m=2,3,5,7 and for the C_{n p^3}(R) case with p odd prime. These outputs will be placed in a new appendix or subsection immediately following the program descriptions. revision: yes
Referee: C++ program presentation: no line-by-line correspondence is given between the code and the formal definitions of Type-1 and Type-2 isomorphism from the referenced earlier papers; in particular, the gcd and connection-set mapping logic is not shown to match the definitions.

Authors: The C++ code was written to implement the Type-1 and Type-2 definitions exactly as stated in the earlier papers of the series (especially the gcd condition m = gcd(n,r) > 1 and the corresponding connection-set mapping). To make this transparent we will insert detailed inline comments at the relevant functions and add a short explanatory subsection that maps the key code blocks (gcd computation, connection-set transformation, and isomorphism check) directly to the formal statements in the cited references. revision: yes
Referee: Verification section (or lack thereof): the manuscript contains no execution trace on the canonical Elspas-Turner example C_16(1,3,7) ≅ C_16(2,3,5), leaving open the possibility of off-by-one or gcd-handling errors that would silently produce spurious families.

Authors: We accept that an explicit verification run on the Elspas-Turner example is necessary. In the revised manuscript we will include a complete execution trace (input, intermediate gcd and mapping steps, and final output) for C_16(1,3,7) and C_16(2,3,5) using the supplied C++ program, thereby confirming that the implementation correctly reproduces the known isomorphism without off-by-one or gcd errors. revision: yes

Circularity Check

1 steps flagged

Minor self-citation for Type-2 definition; computational search results remain independent

specific steps

self citation load bearing [Abstract]
"Vilfred [v96] gave its answer by defining Type-2 isomorphism of C_n(R) w.r.t. m ∋ m = gcd(n, r) > 1, r∈R and r,n∈N and studied such graphs for m = 2 in [v13,v20]."

The operational meaning of 'Type-2 isomorphism' that the C++ program is claimed to detect is supplied entirely by the self-citation [v96]; the present manuscript supplies no independent formalization or verification of that definition inside its own text.

full rationale

The manuscript presents explicit C++ and VB source code that performs exhaustive search over connection sets to detect isomorphisms matching the definitions given in the cited prior papers. No algebraic identity, fitted parameter, or ansatz is derived within this paper that reduces to its own inputs. The only load-bearing reference is the definition of Type-2 isomorphism itself, which is imported from the author's earlier work but functions as an external formal definition rather than a result that is re-proved or fitted here. The concrete families reported are therefore outputs of the supplied programs, not tautological restatements of the input definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard definition of circulant graphs and the Type-2 isomorphism condition introduced in the authors' earlier papers; no new free parameters or invented entities are introduced.

axioms (1)

standard math Standard definition of circulant graph C_n(R) and the Type-2 isomorphism condition w.r.t. m = gcd(n,r) > 1.
Invoked throughout the abstract and program descriptions.

pith-pipeline@v0.9.0 · 5605 in / 1226 out tokens · 47135 ms · 2026-05-15T02:11:18.171518+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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