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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.09491 |
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| _version_ | 1866917399053205504 |
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| author | Roldan, Diego |
| author_facet | Roldan, Diego |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09491 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Graph Energy Maximisation for Integral Circulant Graphs of Order $n = p^2q^3$ Roldan, Diego Combinatorics 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. |
| title | Graph Energy Maximisation for Integral Circulant Graphs of Order $n = p^2q^3$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.09491 |