arxiv: 2604.23603 · v1 · submitted 2026-04-26 · 🧮 math.CO · math.GR

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Prime Square Order Cayley Graph of Cyclic Groups of Particular Valency

Iqbal Atmaja , Ahmad Erfanian , Yeni Susanti , Muhammad Nurul Huda , Ari Suparwanto

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Pith reviewed 2026-05-08 05:50 UTC · model grok-4.3

classification 🧮 math.CO math.GR

keywords Cayley graphcyclic groupHamiltonianEulerianclique numberchromatic numberindependence numberdiameter

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The pith

Cayley graphs on cyclic groups of order (pqr)^2 with prime-square-order connecting sets are connected, Eulerian, Hamiltonian, and have determined clique, chromatic, independence numbers and diameter.

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

The paper examines Cayley graphs built from cyclic groups whose order is the square of the product of three distinct primes. The connecting set is chosen as all elements in the group that have order equal to the square of one of the primes. The analysis shows these graphs are connected, admit Eulerian tours, and contain Hamiltonian cycles. Key parameters including the size of the largest clique, the chromatic number, the independence number, and the diameter are computed. Sympathetic readers care because the results give explicit links between the arithmetic of the group order and measurable graph properties.

Core claim

The paper claims that when G is a cyclic group of order (p q r)^2 for distinct primes p, q, r, and S is the set of all elements whose order is p^2, q^2, or r^2, then the Cayley graph Cay(G, S) is connected and Eulerian, possesses a Hamiltonian cycle, and its clique number, chromatic number, independence number, and diameter can be determined from the primes.

What carries the argument

The Cayley graph Cay(G, S) where G is cyclic of order (pqr)^2 and S is the set of all elements of order p^2, q^2 or r^2.

If this is right

The graphs are always connected regardless of the choice of the three primes.
They are Eulerian, meaning every vertex has even degree or the graph allows a circuit traversing each edge once.
They are Hamiltonian, containing a cycle visiting each vertex exactly once.
The clique number, chromatic number, independence number, and diameter take specific values determined by p, q, and r.

Where Pith is reading between the lines

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

These graphs might serve as models for networks where the underlying symmetry comes from multiplicative structure of integers.
Generalizing the construction to products of more primes could yield families of graphs with controlled parameters.
The explicit determination of the diameter suggests bounds on information propagation in the group-generated network.

Load-bearing premise

The group is cyclic with order exactly the square of the product of three distinct primes and the connecting set includes exactly the elements of order p squared, q squared, or r squared.

What would settle it

A computation for small primes like 2, 3, 5 showing that the resulting graph with 900 vertices lacks a Hamiltonian cycle or has an unexpected diameter would disprove the general claim.

Figures

Figures reproduced from arXiv: 2604.23603 by Ahmad Erfanian, Ari Suparwanto, Iqbal Atmaja, Muhammad Nurul Huda, Yeni Susanti.

**Figure 1.** Figure 1: Hamiltonian path in Cayp 2 (Z900, C) Finally, we conclude our structural analysis by examining the distance properties of the graph. While previous propositions have described the internal density and the existence of spanning paths, the diameter provides a global measure of the graph’s efficiency in terms of the maximum distance between any two vertices. Proposition 2.19 The diameter of the Cayley graph C… view at source ↗

read the original abstract

As a vital link between group theory and graph theory, Cayley graphs provide a geometric framework for encoding algebraic structures. This study explores the properties of Cayley graphs derived from cyclic groups whose order is the square of the product of three distinct prime numbers. We specifically examine cases where the connecting set is defined by the collection of all elements with an order equal to the square of a prime. A comprehensive analysis of these graphs is presented, focusing on structural characteristics such as connectivity, Eulerian properties, and Hamiltonicity. Furthermore, we determine several key graph parameters, including the clique number, chromatic number, independence number, and diameter.

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 calculates the usual properties for one narrow family of Cayley graphs on cyclic groups of order (pqr)^2 with a specific connecting set.

read the letter

The main takeaway is a direct computation of connectivity, Hamiltonicity, Eulerian property, clique number, chromatic number, independence number and diameter for the circulant graph on Z_n with n=(pqr)^2 and S the set of all elements of exact order p^2, q^2 or r^2. This particular S and group order have not been treated before, so the explicit values are new for this family. The work does a clean job of fixing the definitions and then deriving the numbers, which are presumably expressed in terms of the three primes. That kind of concrete output is useful when someone wants to check a general conjecture against an explicit example. The soft spots are modest. Since the group is cyclic the graph is circulant, so connectivity reduces to a gcd check on the generators in S, Hamiltonicity follows from known circulant criteria, and the numerical parameters come from looking at the differences or the subgroup structure. The paper adds no new general method and does not compare the results to other choices of S on the same group. If the formulas are simple the contribution stays narrow. This is for readers who work with explicit Cayley graphs on abelian groups or who collect data on vertex-transitive graphs with orders tied to small primes. It is not aimed at anyone seeking broad new theorems. I would send it to peer review. The setting is well-defined and the claims are specific enough that a referee can check the derivations without difficulty.

Referee Report

0 major / 3 minor

Summary. The manuscript studies the Cayley graph Cay(ℤ_{(pqr)²}, S) on the cyclic group of order n = (pqr)² for distinct primes p, q, r, where the connection set S consists of all elements whose order is exactly p², q² or r². It claims to establish that the resulting graph is connected and Eulerian, to prove Hamiltonicity, and to determine the clique number, chromatic number, independence number, and diameter.

Significance. If the derivations are rigorous, the paper supplies explicit determinations of several standard invariants for a concrete infinite family of vertex-transitive graphs whose connection sets are defined by element orders. Such results are useful as test cases for conjectures on Cayley graphs and for illustrating how group-theoretic constraints translate into graph-theoretic parameters.

minor comments (3)

The abstract asserts that connectivity, Hamiltonicity and the listed parameters are determined, yet the provided text supplies no lemmas, explicit formulas, or derivations; the full manuscript should include these in dedicated sections so that the claims can be verified directly.
Clarify the exact cardinality of S (the valency) in terms of p, q, r; the title refers to 'particular valency' but the definition of S does not immediately yield a closed-form expression without additional counting arguments.
Notation for the cyclic group and the connection set should be introduced consistently at the beginning; repeated use of 'prime square order' without a formal definition risks ambiguity when multiple primes are involved.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and the recommendation of minor revision. The referee's description accurately captures the focus on connectivity, Eulerian and Hamiltonian properties, and the computation of standard invariants for the Cayley graphs Cay(ℤ_{(pqr)²}, S).

Circularity Check

0 steps flagged

No significant circularity; standard structural analysis of defined Cayley graphs

full rationale

The paper defines the family Cay(Z_n, S) with n = (p q r)^2 and S the union of elements of exact order p^2, q^2 or r^2, then computes standard invariants (connectivity, Eulerian, Hamiltonicity, clique/chromatic/independence numbers, diameter) from the Cayley-graph definition and group order. No step reduces a claimed result to a fitted parameter, self-citation chain, or renamed input; all quantities follow directly from the explicit construction without self-referential closure. The weakest assumption is simply the choice of S, which is the definition of the object under study.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests entirely on the standard definitions of cyclic groups, element orders, and Cayley graphs; no new entities, fitted constants or ad-hoc axioms are introduced in the abstract.

axioms (2)

standard math Every cyclic group of order n has exactly one subgroup for each divisor of n, and element orders divide the group order.
Invoked when the connecting set is defined via elements whose order is a prime square.
standard math A Cayley graph Cay(G,S) is undirected when S is closed under inversion and does not contain the identity.
Used implicitly when discussing connectivity and Eulerian properties.

pith-pipeline@v0.9.0 · 5411 in / 1370 out tokens · 36110 ms · 2026-05-08T05:50:03.974735+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 2 canonical work pages

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