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

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 $n=p^2{q}^3$, 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.