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

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.

• The paper establishes an explicit Kronecker factorisation of the adjacency spectrum for a chosen divisor set D in ICGs of order p²q³.
• It derives the first closed-form polynomial formula for the maximal energy, confirmed through extensive computation over 437 prime pairs.
• The study suggests the universality of the unique maximising divisor set, offering implications for optimal graph design and quantum network analysis.

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

Introduction and Background

This paper rigorously addresses the problem of determining the maximum energy of integral circulant graphs (ICGs) of composite order $n = p^2q^3$ for distinct odd primes $p$ and $q$. The graph energy, $E(G)$, is defined as the sum of the absolute values of the adjacency eigenvalues and holds substantial significance in both spectral graph theory and quantum phenomena such as quantum spin networks. While the energy maximisation problem for ICGs of prime-power order has been completely solved in prior works, the extension to bi-prime orders like $p^2q^3$ had remained open due to the more intricate structure of the divisor lattice and the resulting spectrum.

Integral circulant graphs $ICG(n, \mathcal{D})$ are Cayley graphs on the cyclic group $\mathbb{Z}_n$ with connection set determined by a set of divisors $\mathcal{D}$. The spectrum of these graphs, parametrised by Ramanujan sums, admits exact computation due to integrality, enabling algebraic techniques in the search for energy extrema.

Main Results

The central achievement of the study is an explicit Kronecker factorisation of the adjacency spectrum for a particular divisor set $D = \{1, p^2, pq, q^2, p^2q^2, pq^3\}$. Specifically, for every divisor $n = p^2q^3$0 of $n = p^2q^3$1 (excluding $n = p^2q^3$2 itself), the corresponding eigenvalue $n = p^2q^3$3 splits multiplicatively into a function of $n = p^2q^3$4 and a function of $n = p^2q^3$5 for all $n = p^2q^3$6, and is given explicitly for all $n = p^2q^3$7. This structural result (Theorem~1) holds unconditionally for all odd primes.

Based on this algebraic decomposition, the author derives the first closed-form polynomial formula for the maximal energy of an order $n = p^2q^3$8 ICG:

$n = p^2q^3$9

where $p$0 is as specified above.

The paper further provides computational confirmation for all 437 pairs of odd primes with $p$1 that $p$2 is the unique maximising divisor set, leading to the conjecture that this property is universal for all choices of distinct odd primes.

Technical Approach

Ramanujan Sums and Spectral Factorisation

The eigenvalues of an ICG are defined as sums over Ramanujan sums $p$3, and crucially, the multiplicativity of the Ramanujan sum in coprime arguments enables the factorisation of the spectrum for $p$4. By explicit computation, the paper demonstrates that for the chosen set $p$5, the spectrum exhibits exact Kronecker structure—a product of a $p$6-dependent and $p$7-dependent part—for the majority of divisors (all with $p$8). For $p$9, further direct computation yields explicit formulas.

This factorisation explains why the energy for this divisor set can be written as a closed-form polynomial in $q$0 and $q$1, since the summation over all proper divisors in the energy formula becomes tractable.

Closed-Form Energy Derivation

Employing the factorised eigenvalue expressions, the closed-form formula for energy becomes a finite sum over products and direct evaluations at the exponents of $q$2 and $q$3. Crucially, the selection of the maximising set $q$4 ensures all summands are non-negative and polynomially dependent on the prime parameters. The approach fundamentally extends the single-prime exponent minimisation strategy of Sander and Sander, but adapts it by leveraging algebraic, not combinatorial, structure.

Computational Validation

A full enumeration over every non-empty subset $q$5 for all $q$6 was carried out. This systematically establishes that for all tested parameter pairs, the set $q$7 is the unique maximising configuration and the explicit formula matches the computed energies without exception. No other divisor set matches the constructed value.

Implications and Open Problems

The Kronecker factorisation result signals that higher-order structural symmetries may exist for other two-prime or multi-prime cases, hinting that similar explicit spectra may be achievable in those scenarios by identifying "separating" divisor sets. The algebraic proof of universality, presently conjectural beyond the computational base, would likely require generalisation of the convolution and multiplicative frameworks previously used for the prime-power case, but adapted to the more complex divisor lattices of composite bi-prime orders.

If proven, the universality of this maximiser would complete the program of energy maximisation for all integral circulant graphs of order $q$8, extend existing spectral extremal theory, and have direct implications for network design where transmission efficiency (correlating to energy) is paramount.

The singular Kronecker factorisation observed for $q$9 also raises the question of whether these algebraic symmetries are rare or systematically discoverable for larger or more general $E(G)$0 cases. The difficulty lies in the loss of total ordering in the divisor lattice when moving from prime-powers to mixed-prime exponents, which previously facilitated the convexity and combinatorial techniques.

From an applied perspective, the explicit knowledge of maximal energy configurations directly informs the design of optimal quantum networks on such graph topologies. For theoretical chemistry, it provides valuable cases where energy can be exactly computed even in highly composite molecular graphs.

Conclusion

This work solves the energy maximisation problem for integral circulant graphs of order $E(G)$1 by establishing both an unconditional algebraic Kronecker factorisation of the spectrum for a distinguished divisor set and providing the first closed-form polynomial for the maximal energy. Extensive computation supports the universality of this result and positions the factorisation technique as a promising direction for extending energy extremal theories in spectral graph theory. The main open challenge remains the full algebraic proof of uniqueness for the maximising set in all parameter regimes and the extension of these methods to general multi-prime exponents.