Graph Energy Maximisation for Integral Circulant Graphs of Order $n = p^2q^3$

Diego Roldan

arxiv: 2604.09491 · v1 · submitted 2026-04-10 · 🧮 math.CO

Graph Energy Maximisation for Integral Circulant Graphs of Order n = p²q³

Diego Roldan This is my paper

Pith reviewed 2026-05-10 17:13 UTC · model grok-4.3

classification 🧮 math.CO

keywords integral circulant graphsgraph energyeigenvalue factorizationKronecker productdivisor setsenergy maximizationprime powersspectrum separation

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The pith

Eigenvalues of specific integral circulant graphs on p²q³ vertices separate into independent p-only and q-only factors.

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

The paper proves that the adjacency eigenvalues of integral circulant graphs of order p squared times q cubed, using the divisor set consisting of 1, p squared, p times q, q squared, p squared times q squared, and p times q cubed, factor exactly into one component that depends only on the prime p and another that depends only on the prime q. This complete separation holds for every pair of distinct odd primes without further restrictions. The factorization immediately supplies the first closed-form polynomial expression for the energy, defined as the sum of absolute eigenvalues, for any graph in this two-prime family at the given divisor set. Exhaustive checks across many prime pairs show that this particular divisor set produces the highest energy value in each case, and the authors conjecture that the same divisor set is the unique maximizer for all such orders.

Core claim

The adjacency eigenvalues of ICG(p²q³, D*) for D* = {1, p², pq, q², p²q², pq³} admit an exact Kronecker factorisation in the prime exponents: they separate completely into a factor depending only on p and a factor depending only on q. This factorisation holds unconditionally for all pairs of distinct odd primes and constitutes the structural core of the paper. From it we derive, unconditionally, the first closed-form polynomial formula for the energy of a two-prime-order integral circulant graph evaluated at D*. Exhaustive computation over prime pairs (p,q) confirms that D* is the unique energy maximiser in every tested case; we conjecture that this universality holds for all pairs of odd p,

What carries the argument

The exact Kronecker factorisation of the adjacency spectrum of ICG(p²q³, D*) into a p-dependent factor and a q-dependent factor, where D* is the six-element divisor set {1, p², pq, q², p²q², pq³}.

If this is right

The graph energy admits an explicit polynomial formula in the two primes p and q.
The divisor set D* maximises energy among all possible choices in every computationally verified case.
The eigenvalue separation applies without exceptions to all distinct odd primes p and q.
Energy comparisons between different prime pairs become direct algebraic evaluations rather than separate spectral computations.

Where Pith is reading between the lines

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

The same separation technique could be tested on integral circulant graphs whose order is a product of three or more distinct odd primes.
The closed-form energy expression permits direct asymptotic comparison of maximum energies as the primes p and q grow large.
If the maximiser conjecture holds, it singles out one canonical connection pattern that achieves the highest energy for every order of this form.

Load-bearing premise

The chosen divisor set D* produces adjacency eigenvalues that factor exactly into independent contributions from the exponents of p and of q.

What would settle it

A single pair of distinct odd primes p and q for which the eigenvalues of ICG(p²q³, D*) do not factor as a Kronecker product of a p-only term and a q-only term, or for which a different divisor set yields a strictly larger energy.

read the original abstract

The energy of a graph is the sum of the absolute values of its adjacency eigenvalues. For integral circulant graphs $\ICG(n,\mathcal{D})$ of order $n=p^2q^3$, where $p$ and $q$ are distinct odd primes, we prove that the adjacency eigenvalues of $\ICG(p^2q^3,\Dstar)$, for the divisor set $\Dstar=\{1,p^2,pq,q^2,p^2q^2,pq^3\}$, admit an exact Kronecker factorisation in the prime exponents: they separate completely into a factor depending only on $p$ and a factor depending only on~$q$. This factorisation holds unconditionally for all pairs of distinct odd primes and constitutes the structural core of the paper. From it we derive, unconditionally, the first closed-form polynomial formula for the energy of a two-prime-order integral circulant graph evaluated at $\Dstar$. Exhaustive computation over prime pairs $(p,q)$ confirms that $\Dstar$ is the unique energy maximiser in every tested case; we conjecture that this universality holds for all pairs of distinct odd primes.

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

The paper proves an unconditional Kronecker factorization of eigenvalues for ICG(p²q³, D*) that yields a closed-form energy polynomial, but the claim that D* maximizes energy is only supported by checks on small primes.

read the letter

The main result is a proof that the adjacency eigenvalues of this specific integral circulant graph factor completely into a p-dependent part and a q-dependent part. The factorization holds for any distinct odd primes p and q, and it directly produces an explicit polynomial expression for the graph energy. That closed form is new for orders with exactly two distinct odd prime factors and follows from the algebraic structure of circulant graphs without additional assumptions or fitting. The derivation looks clean and uses standard eigenvalue formulas for these graphs, separating the contributions by the prime-power divisors in D* = {1, p², pq, q², p²q², pq³}. This is the part that holds up well and gives a concrete advance in the area. The weaker element is the maximizer statement. Exhaustive checks on small prime pairs show D* always wins and is unique in those cases, but the paper offers no general argument that this pattern continues for larger primes or rules out other divisor sets producing higher energy. The conjecture is reasonable on the evidence given, yet it remains computational rather than proven. The work is aimed at researchers in spectral graph theory who focus on circulant graphs and closed-form energy expressions. A reader interested in factorization techniques or prime-power orders would find the method useful. The math is consistent on its own terms, the citations track relevant prior results on integral circulant graphs, and there are no obvious gaps in the stated claims. I would bring this to a reading group if the group covers algebraic graph theory. It deserves peer review because the core theorem is established and the conjecture is clearly separated with supporting data.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that the adjacency eigenvalues of the integral circulant graph ICG(p²q³, D*) with D*={1,p²,pq,q²,p²q²,pq³} admit an exact Kronecker factorization separating into a p-dependent factor and a q-dependent factor for any distinct odd primes p and q. This factorization is unconditional and yields a closed-form polynomial expression for the graph energy. Exhaustive computational checks over small prime pairs are used to identify D* as the unique energy maximizer in tested cases, with the claim conjectured to hold universally.

Significance. The unconditional factorization and resulting closed-form energy formula constitute a clear algebraic advance for two-prime-order integral circulant graphs, relying on the circulant structure rather than fitted parameters. This provides an exact, non-numerical expression that can be evaluated directly. The computational support for energy maximization is suggestive but remains conjectural; elevating it would strengthen the contribution to graph energy studies.

major comments (1)

[computational verification and conjecture statement] The maximizer claim for D* is supported solely by exhaustive enumeration for small tested prime pairs (p,q), without a general proof, bound, or argument excluding higher-energy divisor sets for larger primes. This renders the central maximization result (central to the title) conjectural rather than established, as the abstract itself acknowledges.

minor comments (2)

[Abstract] The abstract states that exhaustive computation confirms D* is the unique maximizer but does not specify the exact range of primes tested or the full list of divisor sets considered; adding these details would improve reproducibility.
Notation for the divisor set D* and the Kronecker factorization should be introduced with a brief example for concrete small primes (e.g., p=3, q=5) before the general statement to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and for distinguishing the proven algebraic results from the conjectural maximization claim. We respond to the major comment below.

read point-by-point responses

Referee: The maximizer claim for D* is supported solely by exhaustive enumeration for small tested prime pairs (p,q), without a general proof, bound, or argument excluding higher-energy divisor sets for larger primes. This renders the central maximization result (central to the title) conjectural rather than established, as the abstract itself acknowledges.

Authors: We agree that the assertion that D* is the unique energy maximizer remains conjectural. The manuscript provides exhaustive computational verification only for small prime pairs and explicitly states in the abstract that universality is conjectured rather than proved. No general proof, asymptotic bound, or exclusion argument for other divisor sets at large primes is supplied, as none is currently available. The core unconditional results are the Kronecker factorization of the eigenvalues and the resulting closed-form energy polynomial for this specific D*, both of which hold for every pair of distinct odd primes. To align the title more precisely with the established content, we will revise it to 'Closed-Form Energy of the Integral Circulant Graph ICG(p²q³, D*)' and expand the discussion section to clarify the scope of the conjecture and the computational evidence. revision: partial

Circularity Check

0 steps flagged

No circularity: factorization and energy formula derived algebraically from circulant structure

full rationale

The paper proves an unconditional Kronecker factorization of the adjacency eigenvalues of ICG(p²q³, D*) that separates into independent p- and q-dependent factors, using the algebraic definition of integral circulant graphs. The closed-form energy polynomial is then obtained directly by summing the absolute values of these factored eigenvalues. This chain relies on standard circulant eigenvalue formulas and explicit divisor-set arithmetic rather than any fitted parameters, self-definitions, or load-bearing self-citations. The maximizer claim is explicitly labeled a conjecture based on finite enumeration and is not presented as a derived theorem, so it introduces no circular reduction into the proven results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard results about the spectra of circulant graphs and properties of Kronecker products; no free parameters, new axioms, or invented entities are introduced.

axioms (2)

standard math Eigenvalues of circulant graphs are given by sums over roots of unity weighted by the divisor set
This is the standard formula used to obtain the factorization for the chosen D*.
standard math Kronecker product structure preserves eigenvalue products when matrices factor over coprime orders
Invoked to separate the p-exponent and q-exponent contributions.

pith-pipeline@v0.9.0 · 5501 in / 1451 out tokens · 59491 ms · 2026-05-10T17:13:57.693875+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[1]

T. M. Apostol,Introduction to Analytic Number Theory, Springer, New York, 1976

work page 1976
[2]

Baˇ si´ c, M

M. Baˇ si´ c, M. D. Petkovi´ c, D. Stevanovi´ c, Perfect state transfer in integral circulant graphs,Appl. Math. Lett.22(2009), 1117–1121

work page 2009
[3]

Gutman, The energy of a graph,Ber

I. Gutman, The energy of a graph,Ber. Math.-Stat. Sekt. Forschungszent. Graz103 (1978), 1–22

work page 1978
[4]

Ili´ c, The energy of unitary Cayley graphs,Linear Algebra Appl.431(2009), 1881– 1889

A. Ili´ c, The energy of unitary Cayley graphs,Linear Algebra Appl.431(2009), 1881– 1889

work page 2009
[5]

Ili´ c, M

A. Ili´ c, M. Baˇ si´ c, New results on the energy of integral circulant graphs,Appl. Math. Comput.218(2011), 3470–3482

work page 2011
[6]

Kiani, M

D. Kiani, M. M. H. Aghaei, Y. Meemark, B. Suntornpoch, Energy of unitary Cayley graphs and gcd-graphs,Linear Algebra Appl.435(2011), 1336–1343

work page 2011
[7]

Klotz, T

W. Klotz, T. Sander, Some properties of unitary Cayley graphs,Electron. J. Combin. 14(2007), R45

work page 2007
[8]

Klotz, T

W. Klotz, T. Sander, GCD-graphs and NEPS of complete graphs,Ars Math. Con- temp.6(2013), 289–299

work page 2013
[9]

T. A. Le, J. W. Sander, Extremal energies of integral circulant graphs via multiplica- tivity,Linear Algebra Appl.437(2012), 1408–1421

work page 2012
[10]

T. A. Le, J. W. Sander, Convolutions of Ramanujan sums and integral circulant graphs,Int. J. Number Theory8(2012), 1777–1788

work page 2012
[11]

X. Li, Y. Shi, I. Gutman,Graph Energy, Springer, New York, 2012

work page 2012
[12]

M. D. Petkovi´ c, M. Baˇ si´ c, Further results on the perfect state transfer in integral circulant graphs,Comput. Math. Appl.61(2011), 300–312

work page 2011
[13]

D. G. Rold´ an, Graph energy maximisation for integral circulant graphs of orderpaqb: first exact results — the casep 3q2,Discrete Appl. Math., submitted (2026)

work page 2026
[14]

J. W. Sander, Integral circulant Ramanujan graphs via multiplicativity and ultrafri- able integers,Linear Algebra Appl.477(2015), 21–41

work page 2015
[15]

J. W. Sander, T. Sander, The energy of integral circulant graphs with prime power order,Appl. Anal. Discrete Math.5(2011), 22–36. 7

work page 2011
[16]

J. W. Sander, T. Sander, Integral circulant graphs of prime power order with maximal energy,Linear Algebra Appl.435(2011), 3212–3232

work page 2011
[17]

J. W. Sander, T. Sander, The maximal energy of classes of integral circulant graphs, Discrete Appl. Math.160(2012), 2015–2029

work page 2012
[18]

J. W. Sander, T. Sander, The exact maximal energy of integral circulant graphs with prime power order,Contrib. Discrete Math.8(2013), 19–40

work page 2013
[19]

J. W. Sander, T. Sander, On So’s conjecture for integral circulant graphs,Appl. Anal. Discrete Math.9(2015), 59–72

work page 2015
[20]

Saxena, S

N. Saxena, S. Severini, I. E. Shparlinski, Parameters of integral circulant graphs and periodic quantum dynamics,Int. J. Quant. Inf.5(2007), 417–430

work page 2007
[21]

So, Integral circulant graphs,Discrete Math.306(2006), 153–158

W. So, Integral circulant graphs,Discrete Math.306(2006), 153–158. 8

work page 2006

[1] [1]

T. M. Apostol,Introduction to Analytic Number Theory, Springer, New York, 1976

work page 1976

[2] [2]

Baˇ si´ c, M

M. Baˇ si´ c, M. D. Petkovi´ c, D. Stevanovi´ c, Perfect state transfer in integral circulant graphs,Appl. Math. Lett.22(2009), 1117–1121

work page 2009

[3] [3]

Gutman, The energy of a graph,Ber

I. Gutman, The energy of a graph,Ber. Math.-Stat. Sekt. Forschungszent. Graz103 (1978), 1–22

work page 1978

[4] [4]

Ili´ c, The energy of unitary Cayley graphs,Linear Algebra Appl.431(2009), 1881– 1889

A. Ili´ c, The energy of unitary Cayley graphs,Linear Algebra Appl.431(2009), 1881– 1889

work page 2009

[5] [5]

Ili´ c, M

A. Ili´ c, M. Baˇ si´ c, New results on the energy of integral circulant graphs,Appl. Math. Comput.218(2011), 3470–3482

work page 2011

[6] [6]

Kiani, M

D. Kiani, M. M. H. Aghaei, Y. Meemark, B. Suntornpoch, Energy of unitary Cayley graphs and gcd-graphs,Linear Algebra Appl.435(2011), 1336–1343

work page 2011

[7] [7]

Klotz, T

W. Klotz, T. Sander, Some properties of unitary Cayley graphs,Electron. J. Combin. 14(2007), R45

work page 2007

[8] [8]

Klotz, T

W. Klotz, T. Sander, GCD-graphs and NEPS of complete graphs,Ars Math. Con- temp.6(2013), 289–299

work page 2013

[9] [9]

T. A. Le, J. W. Sander, Extremal energies of integral circulant graphs via multiplica- tivity,Linear Algebra Appl.437(2012), 1408–1421

work page 2012

[10] [10]

T. A. Le, J. W. Sander, Convolutions of Ramanujan sums and integral circulant graphs,Int. J. Number Theory8(2012), 1777–1788

work page 2012

[11] [11]

X. Li, Y. Shi, I. Gutman,Graph Energy, Springer, New York, 2012

work page 2012

[12] [12]

M. D. Petkovi´ c, M. Baˇ si´ c, Further results on the perfect state transfer in integral circulant graphs,Comput. Math. Appl.61(2011), 300–312

work page 2011

[13] [13]

D. G. Rold´ an, Graph energy maximisation for integral circulant graphs of orderpaqb: first exact results — the casep 3q2,Discrete Appl. Math., submitted (2026)

work page 2026

[14] [14]

J. W. Sander, Integral circulant Ramanujan graphs via multiplicativity and ultrafri- able integers,Linear Algebra Appl.477(2015), 21–41

work page 2015

[15] [15]

J. W. Sander, T. Sander, The energy of integral circulant graphs with prime power order,Appl. Anal. Discrete Math.5(2011), 22–36. 7

work page 2011

[16] [16]

J. W. Sander, T. Sander, Integral circulant graphs of prime power order with maximal energy,Linear Algebra Appl.435(2011), 3212–3232

work page 2011

[17] [17]

J. W. Sander, T. Sander, The maximal energy of classes of integral circulant graphs, Discrete Appl. Math.160(2012), 2015–2029

work page 2012

[18] [18]

J. W. Sander, T. Sander, The exact maximal energy of integral circulant graphs with prime power order,Contrib. Discrete Math.8(2013), 19–40

work page 2013

[19] [19]

J. W. Sander, T. Sander, On So’s conjecture for integral circulant graphs,Appl. Anal. Discrete Math.9(2015), 59–72

work page 2015

[20] [20]

Saxena, S

N. Saxena, S. Severini, I. E. Shparlinski, Parameters of integral circulant graphs and periodic quantum dynamics,Int. J. Quant. Inf.5(2007), 417–430

work page 2007

[21] [21]

So, Integral circulant graphs,Discrete Math.306(2006), 153–158

W. So, Integral circulant graphs,Discrete Math.306(2006), 153–158. 8

work page 2006