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arxiv: 2211.06970 · v2 · pith:6WCBZ7NTnew · submitted 2022-11-13 · 🧮 math.CO

A study on Type-2 isomorphic circulant graphs. Part 10: Type-2 isomorphic C_(np³)(R) w.r.t. m = p and related groups

Vilfred Kamalappan , Wilson Peraprakash This is my paper

Pith reviewed 2026-05-24 10:41 UTC · model grok-4.3

classification 🧮 math.CO
keywords circulant graphsType-2 isomorphismAbelian groupsconnection setsprime pgraph isomorphism
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The pith

Circulant graphs C_{np^3}(R^{np^3,x+yp}_i) for i=1 to p are Type-2 isomorphic wrt m=p and form Abelian groups of order p.

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

The paper proves that when the connection set R^{np^3,x+yp}_i contains both p and np^3-p and satisfies the stated bounds on x and y, the p graphs obtained by varying the index i from 1 to p are pairwise Type-2 isomorphic with respect to the parameter m equal to the prime p. These graphs are closed under the operation that adds an integer j to the index i modulo p, and the resulting set equipped with this operation forms an Abelian group denoted T2_{np^3,p}(C_{np^3}(R^{np^3,x+yp}_i)). Explicit lists of such groups are supplied for small primes p=3,5,7 and small n, with further families indicated in related parts of the series.

Core claim

For each admissible x and y, the p circulant graphs C_{np^3}(R^{np^3,x+yp}_i) with i=1 to p are Type-2 isomorphic with respect to m=p; the collection of these graphs under the index-shift operation forms the Abelian group T2_{np^3,p}(C_{np^3}(R^{np^3,x+yp}_i)) whose elements are exactly the images under the maps θ_{np^3,p,jn}.

What carries the argument

The index-shift isomorphism θ_{np^3,p,jn} that sends C_{np^3}(R^{np^3,x+yp}_i) to C_{np^3}(R^{np^3,x+yp}_{i+j mod p}), which is shown to be a Type-2 isomorphism when m=p and to close the family into a group.

If this is right

  • Each such family contains exactly p distinct but Type-2-isomorphic circulant graphs.
  • The operation of index addition modulo p is associative and commutative on these sets.
  • Concrete lists of the groups exist for every prime p and every admissible n and y.

Where Pith is reading between the lines

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

  • The construction supplies an explicit way to partition certain families of circulant graphs into orbits of size p under Type-2 isomorphism.
  • If the same index-shift closure holds when m is composite, the same proof technique would produce groups of order m rather than p.

Load-bearing premise

The connection sets are required to contain both p and np^3-p (and obey the listed bounds on x and y) so that every index shift modulo p produces another valid member of the same family.

What would settle it

For p=3, n=1, x=1, y=0, compute the two graphs C_27(R^{27,1}_1) and C_27(R^{27,1}_2) and check whether any Type-2 isomorphism with m=3 exists between them; if none does, the claimed family collapses.

read the original abstract

This study is the $10^{th}$ part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10}. In this part, we obtain families of Type-2 isomorphic circulant graphs $C_{np^3}(R)$ w.r.t. $m$ = $p$, and related Abelian groups where $p$ is a prime number and $n\in\mathbb{N}$. In its main theorem, it is proved that for $i$ = 1 to $p$, circulant graphs $C_{np^3}(R^{np^3,x+yp}_i)$ are isomorphic of Type-2 w.r.t. $m$ = $p$ and they form Abelian group $(T2_{np^3,p}(C_{np^3}(R^{np^3,x+yp}_i)), \circ)$ where $T2_{np^3,p}(C_{np^3}(R^{np^3,x+yp}_i))$ = $\{\theta_{np^3,p,jn}(C_{np^3}(R^{np^3,x+yp}_i))$ = $C_{np^3}(R^{np^3,x+yp}_{i+j}) :$ $j$ = $0,1,...,p-1$ and $i+j$ in $C_{np^3}(R^{np^3,x+yp}_{i+j})$ is calculated under addition modulo $p \}$, $1 \leq x \leq p-1$, $0 \leq y \leq np - 1$, $1 \leq x+yp \leq np^2-1$, $y\in\mathbb{N}_0$, $p,np^3-p\in R^{np^3,x+yp}_i$ and $i,n,x\in\mathbb{N}$. And using it, a list of $T2_{np^3,p}(C_{np^3}(R^{np^3,x+yp}_i))$, each containing $p$ isomorphic circulant graphs $C_{np^3}(R^{np^3,x+yp}_i)$ of Type-2 w.r.t. $m$ = $p$, for $p$ = 3,5,7, $n$ = 1,2 and $y$ = 0 is given in the Annexure and more such families of Type-2 isomorphic circulant graphs are presented in \cite{v24}.

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

3 major / 1 minor

Summary. The paper, the tenth in a series, claims that for prime p and natural n, under the conditions that the connection sets R^{np^3,x+yp}_i contain both p and np^3-p and satisfy the stated bounds on x and y, the circulant graphs C_{np^3}(R^{np^3,x+yp}_i) for i=1 to p are pairwise Type-2 isomorphic with respect to m=p via the index-shift maps θ_{np^3,p,jn}, and that these p graphs form an Abelian group T2_{np^3,p} under an induced operation ∘. Concrete lists of such families are supplied for p=3,5,7 and small n,y in an annexure.

Significance. If the central claim holds, the work supplies explicit, infinite families of Type-2 isomorphic circulant graphs on np^3 vertices together with associated Abelian groups of order p; the annexure lists for small primes give immediately usable examples. This extends the authors' prior installments by furnishing concrete group structures rather than isolated isomorphisms.

major comments (3)
  1. [Abstract] Abstract (main theorem statement): the set T2_{np^3,p}(C_{np^3}(R^{np^3,x+yp}_i)) is defined to be exactly the collection of graphs obtained by applying the maps θ that the theorem asserts are Type-2 isomorphisms; this renders the group construction circular, as membership in T2 presupposes the very isomorphisms the theorem is required to establish independently.
  2. [Abstract] Abstract (main theorem statement): the binary operation ∘ on T2 is never defined; the text only asserts that (T2, ∘) is Abelian without exhibiting the operation, verifying closure under the given conditions on R, or checking the group axioms.
  3. [Abstract] Abstract (main theorem statement): no proof, outline, or reference to a specific lemma from parts 1–9 is supplied for the claim that the index shifts θ preserve the connection-set conditions and induce Type-2 isomorphisms; only the statement and small-case lists appear, so the load-bearing step cannot be checked.
minor comments (1)
  1. [Abstract] The notation R^{np^3,x+yp}_i is introduced without an explicit recursive or set-theoretic definition in the abstract; a forward reference to the definition used in earlier parts would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and detailed review of our manuscript. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (main theorem statement): the set T2_{np^3,p}(C_{np^3}(R^{np^3,x+yp}_i)) is defined to be exactly the collection of graphs obtained by applying the maps θ that the theorem asserts are Type-2 isomorphisms; this renders the group construction circular, as membership in T2 presupposes the very isomorphisms the theorem is required to establish independently.

    Authors: We agree that the abstract wording risks appearing circular. The manuscript first defines the family of graphs via the connection sets R^{np^3,x+yp}_i satisfying the listed conditions (including p and np^3-p belonging to each R_i), independently of the maps θ. The main theorem then proves that the index-shift maps θ_{np^3,p,jn} are Type-2 isomorphisms with respect to m=p. We will revise the abstract to separate the definition of the family from the isomorphism claim. revision: yes

  2. Referee: [Abstract] Abstract (main theorem statement): the binary operation ∘ on T2 is never defined; the text only asserts that (T2, ∘) is Abelian without exhibiting the operation, verifying closure under the given conditions on R, or checking the group axioms.

    Authors: The operation ∘ is the one induced by addition of the indices modulo p, as indicated by the explicit construction of the set T2 via i+j taken modulo p. This ensures closure by definition and commutativity. We acknowledge that an explicit statement of the operation together with a verification of the group axioms under the given bounds on x, y and the membership conditions on R should appear. We will add this definition and a brief verification to the revised abstract and body. revision: yes

  3. Referee: [Abstract] Abstract (main theorem statement): no proof, outline, or reference to a specific lemma from parts 1–9 is supplied for the claim that the index shifts θ preserve the connection-set conditions and induce Type-2 isomorphisms; only the statement and small-case lists appear, so the load-bearing step cannot be checked.

    Authors: The abstract summarizes the result; the proof in the body applies the preservation lemmas for index shifts established in parts 1–9 of the series to the specific families R^{np^3,x+yp}_i. We will insert explicit citations to the relevant lemmas from the earlier parts both in the abstract and at the appropriate place in the body to make the dependence clear. revision: yes

Circularity Check

1 steps flagged

Group T2 defined directly via the θ maps asserted to be isomorphisms

specific steps
  1. self definitional [Abstract / main theorem statement]
    "it is proved that for i = 1 to p, circulant graphs C_{np^3}(R^{np^3,x+yp}_i) are isomorphic of Type-2 w.r.t. m = p and they form Abelian group (T2_{np^3,p}(C_{np^3}(R^{np^3,x+yp}_i)), ∘) where T2_{np^3,p}(C_{np^3}(R^{np^3,x+yp}_i)) = {θ_{np^3,p,jn}(C_{np^3}(R^{np^3,x+yp}_i)) = C_{np^3}(R^{np^3,x+yp}_{i+j}) : j = 0,1,...,p-1 and i+j calculated under addition modulo p}"

    T2 is defined to be the collection of all images under the θ maps; the theorem then claims these images are isomorphic and that the collection forms a group under the induced operation. The group property therefore holds by construction of the set once the R_i are stipulated to contain p and np³-p (ensuring the shifted index remains inside the family).

full rationale

The central theorem asserts that the listed graphs are Type-2 isomorphic and simultaneously defines the set T2 as exactly the orbit under those same θ maps; closure and the group operation are therefore built into the set definition once the connection-set conditions (p, np³-p ∈ R) are granted. This matches self-definitional circularity. The ten-part self-citation chain supplies the prior definition of 'Type-2 isomorphism' but is not itself load-bearing for the algebraic closure step inside this paper. No other patterns (fitted predictions, uniqueness theorems, ansatz smuggling) appear in the supplied text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claim rests on the authors' prior definition of Type-2 isomorphism (from the cited series) together with standard facts about circulant graphs and Abelian groups; the connection-set conditions on R are domain assumptions required for the index-shift construction to close.

axioms (2)
  • standard math Standard definitions and properties of circulant graphs and graph isomorphism
    Invoked throughout the statement of the main theorem.
  • domain assumption The connection sets R always contain p and np^3-p under the stated bounds on x,y
    Listed as a prerequisite in the main theorem conditions.
invented entities (2)
  • Type-2 isomorphism w.r.t. m no independent evidence
    purpose: A specialized equivalence relation on circulant graphs parameterized by m
    Introduced and used in the authors' series; no independent evidence supplied here.
  • T2_{np^3,p} Abelian group no independent evidence
    purpose: The group whose elements are the p isomorphic graphs and whose operation is index addition modulo p
    Constructed directly from the isomorphism relation proved in the theorem.

pith-pipeline@v0.9.0 · 6043 in / 1550 out tokens · 38214 ms · 2026-05-24T10:41:16.522554+00:00 · methodology

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Reference graph

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