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arxiv: 2606.20984 · v1 · pith:YTMHTVHWnew · submitted 2026-06-18 · 🧮 math.CO

A study on Type-2 isomorphic circulant graphs. Part 7: Isomorphism series, digraph and graph of C_n(R)

V. Vilfred Kamalappan This is my paper

Pith reviewed 2026-06-26 16:14 UTC · model grok-4.3

classification 🧮 math.CO
keywords circulant graphsisomorphismType-2 isomorphismisomorphism digraphisomorphism graphisomorphic setC_n(R)
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The pith

Circulant graphs admit isomorphism digraphs and graphs that organize their Type-2 isomorphisms and identify new classes.

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

This paper, the seventh in a series, defines isomorphic sets, isomorphism series, isomorphism digraphs or diagrams, and isomorphism graphs for circulant graphs C_n(R). These are constructed explicitly for small cases like C_16(R), C_27(S), and C_54(1,3,17,19), and illustrated for the larger C_432 with its multiple Type-2 isomorphisms under different m values. The work also demonstrates specific pairs of C_54 graphs that are isomorphic without fitting the Type-1 or Type-2 definitions relative to m=3. It further introduces the diameter of an isomorphic set and the isomorphic distance between two such graphs. These definitions aim to provide a structured way to study the relations among different connection sets that yield isomorphic graphs.

Core claim

We define the isomorphic set of C_n(R) as the collection of all connection sets that give isomorphic circulant graphs, the isomorphism series as ordered sequences within these, the isomorphism digraph D as the directed graph with vertices as distinct graphs and arcs indicating isomorphism relations, and the isomorphism graph G as its undirected counterpart. For the listed examples these objects are determined, and pairs such as C_54(1,3,17,19) with C_54(5,13,21,23) are shown to be isomorphic yet outside the Type-1 and Type-2 categories for m=3. The diameter of the isomorphic set and isomorphic distances are also computed for some cases.

What carries the argument

The isomorphism digraph D and isomorphism graph G, which represent the relations among isomorphic circulant graphs with different connection sets R.

Load-bearing premise

The new isomorphism digraph and graph structures capture distinct and useful information about the isomorphisms that goes beyond merely determining whether two graphs are isomorphic or not.

What would settle it

A direct computation of all isomorphisms among the connection sets for C_54 would confirm or refute whether the identified pairs are isomorphic and whether any others exist that fit or do not fit the Type classifications.

Figures

Figures reproduced from arXiv: 2606.20984 by V. Vilfred Kamalappan.

Figure 1
Figure 1. Figure 1: Digraph of Hamiltonian isomorphism series of [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 6
Figure 6. Figure 6: Digraph of Hamiltonian isomorphism series of [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Digraph D with [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: Digraph D with [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: Digraph D with [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: Digraph D with [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: Digraph D with R1 = {1, 3, 8, 10}, S1 = {3, 4, 5, 13}, T1 = {2, 3, 7, 11}, R2 = {2, 6, 7, 11}, S2 = {1, 6, 8, 10}, T2 = {4, 5, 6, 13}, R3 = {4, 5, 12, 13}, S3 = {2, 7, 11, 12}, T3 = {1, 8, 10, 12}. (2) Here, C27(R) = C27(1, 6, 8, 10), T227,3(C27(1, 6, 8, 10)) = {C27(1, 6, 8, 10), C27(4, 5, 6, 13) = T227,3,1(C27(1, 6, 8, 10)), C27(2, 6, 7, 11) = T227,3,2(C27(1, 6, 8, 10))} = T227,3(C27(4, 5, 6, 13)) = T227… view at source ↗
Figure 16
Figure 16. Figure 16: Graph G with R1 = {1, 3, 8, 10}, S1 = {3, 4, 5, 13}, T1 = {2, 3, 7, 11}, R2 = {2, 6, 7, 11}, S2 = {1, 6, 8, 10}, T2 = {4, 5, 6, 13}, R3 = {4, 5, 12, 13}, S3 = {2, 7, 11, 12}, T3 = {1, 8, 10, 12}. (3) Here, C27(R) = C27(1, 8, 10, 12), C27(1, 8, 10, 12) ∼=T227,3,1 C27(2, 7, 11, 12) and C27(1, 8, 10, 12) ∼=T227,3,2 C27(4, 5, 12, 13). From (1) and (2), we have V (D) = V (G) = Iso(C27(1, 8, 10, 12)) = Iso(C27(… view at source ↗
Figure 18
Figure 18. Figure 18: Graph [PITH_FULL_IMAGE:figures/full_fig_p026_18.png] view at source ↗
Figure 20
Figure 20. Figure 20: Graph [PITH_FULL_IMAGE:figures/full_fig_p028_20.png] view at source ↗
Figure 22
Figure 22. Figure 22: Graph [PITH_FULL_IMAGE:figures/full_fig_p030_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Digraph of Hamiltonian isomorphism series of [PITH_FULL_IMAGE:figures/full_fig_p035_23.png] view at source ↗
Figure 25
Figure 25. Figure 25: Graph [PITH_FULL_IMAGE:figures/full_fig_p037_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Digraph of Hamiltonian isomorphism series of [PITH_FULL_IMAGE:figures/full_fig_p038_26.png] view at source ↗
read the original abstract

This study is the $7^{th}$ part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10}. In this study, we define {\em isomorphic set}, {\em isomorphism series}, {\em isomorphism digraph} $\mathcal{D}$ or {\em isomorphism diagram} and {\em isomorphism graph} $\mathcal{G}$ of circulant graphs and obtain these corresponding to $C_{16}(R)$, $C_{27}(S)$ and $C_{54}(1,3,17,19)$ and present the isomorphism digraph and the isomorphism graph of $C_{432}(16, 27, 48, 54, 128, 160, 189)$ which has isomorphic circulant graphs of Type-2 w.r.t. $m$ = 2 as well as $m$ = 3. We also show that each pair of circulant graphs $C_{54}(1,3,17,19)$, $C_{54}(5,13,21,23)$; $C_{54}(7, 11, 21, 25)$, $C_{54}(7, 11, 15, 25)$; and $C_{54}(1,3,17,19)$, $C_{54}(7,11,15,25)$ are isomorphic but they are neither of Type-1 nor of Type-2 w.r.t. $m$ = 3. More such circulant graphs are given in the conclusion. We also define {\em diameter of isomorphic set} of $C_n(R)$ and {\em isomorphic distance} of $C_n(S)$ and $C_n(T)$ and obtained these values for some circulant graphs.

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

1 major / 2 minor

Summary. The paper, the seventh in a ten-part series on Type-2 isomorphic circulant graphs, defines the isomorphic set, isomorphism series, isomorphism digraph D (or diagram), and isomorphism graph G of circulant graphs C_n(R). It constructs explicit instances of these objects for C_16(R), C_27(S), and C_54(1,3,17,19), presents D and G for the larger example C_432(16,27,48,54,128,160,189) exhibiting Type-2 isomorphisms for both m=2 and m=3, identifies three pairs of C_54 graphs that are isomorphic yet neither Type-1 nor Type-2 with respect to m=3, and introduces the diameter of an isomorphic set together with the isomorphic distance between two such graphs, supplying numerical values for selected examples.

Significance. If the constructions are correct, the auxiliary structures supply diagrammatic and graph-theoretic representations that organize the isomorphism classes of circulant graphs in a manner extending the author's prior Type-1/Type-2 classification. The explicit identification of isomorphisms on 54 vertices lying outside that classification, together with the larger n=432 example, supplies concrete data that could be used to test the completeness of existing typologies. The work is incremental within the series and offers no comparison with standard isomorphism algorithms or external benchmarks.

major comments (1)
  1. [Abstract] Abstract: the assertion that the three listed pairs of C_54 graphs are isomorphic yet neither Type-1 nor Type-2 with respect to m=3 is stated without any derivation, computational procedure, or verification method; this claim is load-bearing for the paper's suggestion that the Type-1/Type-2 framework is incomplete.
minor comments (2)
  1. The manuscript does not restate the definitions of Type-1 and Type-2 isomorphisms from the cited prior parts, forcing readers to consult the series to evaluate the new claims.
  2. Notation for the connection sets (R, S, etc.) and the parameter m is used without an introductory example that would make the constructions self-contained.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the review and for highlighting the need for clearer support of the key claim in the abstract. We address the comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that the three listed pairs of C_54 graphs are isomorphic yet neither Type-1 nor Type-2 with respect to m=3 is stated without any derivation, computational procedure, or verification method; this claim is load-bearing for the paper's suggestion that the Type-1/Type-2 framework is incomplete.

    Authors: The abstract is a concise summary; the explicit verification appears in the body via the newly defined isomorphism series, digraph D, and graph G. For the three C_54 pairs we construct the isomorphic sets, compute the directed edges of D and the edges of G, and confirm both the isomorphisms and that none of the connecting paths correspond to the Type-1 or Type-2 relations previously defined for m=3. These constructions are carried out directly from the connection sets and are therefore self-contained within the manuscript. If the presentation of the verification steps is judged insufficiently prominent, we are prepared to insert a short cross-reference to the relevant sections inside the abstract itself. revision: partial

Circularity Check

0 steps flagged

New definitions and explicit constructions are self-contained

full rationale

The manuscript introduces fresh auxiliary structures (isomorphic set, isomorphism series, digraph D, graph G, diameter, isomorphic distance) and applies them directly to concrete circulant graphs C_16(R), C_27(S), C_54(1,3,17,19) and C_432(...). These constructions are presented as outputs of the new definitions themselves. References to the author's prior parts supply only the baseline Type-1/Type-2 classification against which the new examples are contrasted; the paper does not derive any claimed isomorphism or diagram from those prior definitions by algebraic reduction or fitting. No equation or result inside the manuscript is shown to equal its input by construction, and the explicit pairs on n=54 are asserted to lie outside the prior classification without the classification itself being re-derived here. The work is therefore self-contained as a definitional and enumerative extension.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 4 invented entities

The central claims rest on standard background definitions of circulant graphs plus the Type-1/Type-2 framework from the author's earlier self-citations; several new entities are introduced without independent evidence.

axioms (2)
  • standard math Standard definition and basic properties of circulant graphs C_n(R)
    Invoked implicitly throughout the abstract as the objects under study.
  • domain assumption Validity and completeness of the Type-1 and Type-2 isomorphism classifications from parts 1-6
    The abstract repeatedly contrasts new findings against these prior categories.
invented entities (4)
  • isomorphic set of C_n(R) no independent evidence
    purpose: Group of mutually isomorphic circulant graphs
    Newly defined term whose properties are studied in the paper.
  • isomorphism series of C_n(R) no independent evidence
    purpose: Sequence describing successive isomorphisms
    Newly defined term.
  • isomorphism digraph D no independent evidence
    purpose: Directed graph representing isomorphism relations
    Newly defined structure.
  • isomorphism graph G no independent evidence
    purpose: Undirected graph representing isomorphism relations
    Newly defined structure.

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discussion (0)

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Reference graph

Works this paper leans on

21 extracted references · 3 canonical work pages · 1 internal anchor

  1. [1]

    Adam,Research problem 2-10, J

    A. Adam,Research problem 2-10, J. Combinatorial Theory,3(1967), 393

    1967

  2. [2]

    J. A. Bondy and U. S. R. Murty,Graph Theory with Applications,5 th Edi., Elsevier Sci. Publ. Co., New York, 1982

    1982

  3. [3]

    P. J. Davis,Circulant Matrices,Wiley, New York, 1979

    1979

  4. [4]

    West,Introduction to Graph Theory,2 ed Edi., Pearson Education (Singapore) Pvt

    Dauglas B. West,Introduction to Graph Theory,2 ed Edi., Pearson Education (Singapore) Pvt. Ltd., 2002

    2002

  5. [5]

    Elspas and J

    B. Elspas and J. Turner,Graphs with circulant adjacency matrices, J. Combinatorial Theory,9(1970), 297-307

    1970

  6. [6]

    Kra and S

    I. Kra and S. R. Simanca,On Circulant Matrices, AMS Notices,59(2012), 368–377

    2012

  7. [7]

    Vilfred, ∑ -labelled Graphs and Circulant Graphs, Ph.D

    V. Vilfred, ∑ -labelled Graphs and Circulant Graphs, Ph.D. Thesis, University of Kerala, Thiruvananthapuram, Kerala, India (1996)

    1996

  8. [8]

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

    V. Vilfred Kamalappan,All Type-2 Isomorphic Circulant Graphs ofC 16(R)andC 24(S), arXiv: 2508.09384v1 [math.CO] (12 Aug 2025), 28 pages

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  9. [9]

    Vilfred,A study on Type-2 Isomorphic Circulant Graphs and related Abelian Groups, arXiv: 2012.11372v11 [math.CO] (26 Nov

    V. Vilfred,A study on Type-2 Isomorphic Circulant Graphs and related Abelian Groups, arXiv: 2012.11372v11 [math.CO] (26 Nov. 2024), 183 pages

    work page arXiv 2012

  10. [10]

    Vilfred,A Theory of Cartesian Product and Factorization of Circulant Graphs, Hindawi Pub

    V. Vilfred,A Theory of Cartesian Product and Factorization of Circulant Graphs, Hindawi Pub. Corp. - J. Discrete Math.,V ol. 2013, Article ID 163740, 10 pages

    2013

  11. [11]

    Vilfred Kamalappan,New Families of Circulant Graphs Without Cayley Isomorphism Property withr i = 2, Int

    V. Vilfred Kamalappan,New Families of Circulant Graphs Without Cayley Isomorphism Property withr i = 2, Int. Journal of Applied and Computational Mathematics, (2020) 6:90, 34 pages. https://doi.org/10.1007/s40819-020- 00835-0. Published online: 28.07.2020 by Springer

    work page doi:10.1007/s40819-020- 2020

  12. [12]

    Vilfred Kamalappan,A study on Type-2 Isomorphic Circulant Graphs

    V. Vilfred Kamalappan,A study on Type-2 Isomorphic Circulant Graphs. Part 1: Type-2 isomorphic circulant graphsC n(R)w.r.t.m= 2. Preprint. 31 pages

  13. [13]

    Vilfred Kamalappan,A study on Type-2 isomorphic circulant graphs

    V. Vilfred Kamalappan,A study on Type-2 isomorphic circulant graphs. Part 2: Type-2 isomorphic circulant graphs of orders 16, 24, 27. Preprint. 32 pages

  14. [14]

    Vilfred Kamalappan,A study on Type-2 isomorphic circulant graphs

    V. Vilfred Kamalappan,A study on Type-2 isomorphic circulant graphs. Part 3: 384 pairs of Type-2 isomorphic circulant graphsC 32(R). Preprint. 42 pages

  15. [15]

    Vilfred Kamalappan,A study on Type-2 isomorphic circulant graphs

    V. Vilfred Kamalappan,A study on Type-2 isomorphic circulant graphs. Part 4: 960 triples of Type-2 isomorphic circulant graphsC 54(R). Preprint. 76 pages

  16. [16]

    Vilfred Kamalappan,A study on Type-2 isomorphic circulant graphs

    V. Vilfred Kamalappan,A study on Type-2 isomorphic circulant graphs. Part 5: Type-2 isomorphic circulant graphs of orders 48, 81, 96. Preprint. 33 pages

  17. [17]

    Vilfred Kamalappan,A study on Type-2 Isomorphic Circulant Graphs

    V. Vilfred Kamalappan,A study on Type-2 Isomorphic Circulant Graphs. Part 6: Abelian groups(T2 n,m(Cn(R)),◦)and(Vn,m(Cn(R)),◦). Preprint. 19 pages

  18. [18]

    Vilfred Kamalappan,A study on Type-2 Isomorphic Circulant Graphs

    V. Vilfred Kamalappan,A study on Type-2 Isomorphic Circulant Graphs. Part 7: Isomorphism series, digraph and graph ofC n(R). Preprint. 54 pages

  19. [19]

    Vilfred Kamalappan,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

    V. Vilfred Kamalappan,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. Preprint. 99 pages

  20. [20]

    Vilfred Kamalappan and P

    V. Vilfred Kamalappan and P. Wilson,A study on Type-2 Isomorphic Circulant Graphs. Part 9: Computer program to show Type-1 and -2 isomorphic circulant graphs. Preprint. 21 pages

  21. [21]

    Vilfred Kamalappan and P

    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 A STUDY ON TYPE-2 CIRCULANT GRAPHS: PART 7: ISOMORPHIS...