Energy Maximisation for Integral Circulant Graphs with Opposite-Parity Exponents

For a finite graph, its energy is the sum of the absolute values of its adjacency eigenvalues. Let p and q be distinct odd primes with q at least 5, and let r be at least 1 and s be nonnegative. We determine the maximum energy among all integral circulant graphs whose order is p to the power 2r times q to the power 2s plus 1. The unique energy-maximising divisor set is the checkerboard set consisting of all divisors p raised to the power i times q raised to the power j, where i ranges from zero to 2r, j ranges from zero to 2s plus 1, and i plus j is even, and we obtain an explicit closed formula for the corresponding maximum energy. In particular, for q at least 5, when r and s are both equal to 1, our theorem establishes the conjectured maximality of Roldan's checkerboard divisor set and recovers his closed-form energy formula. The main ingredient is a semidefinite parity theorem for weighted prime-power Ramanujan transforms, proved by a parity-independent congruence reduction and a block Schur recurrence. Centring the divisor matrix then yields a sharp sign-matrix inequality from semidefinite bounds for a Kronecker-product operator. Analysing equality identifies the checkerboard pattern and proves uniqueness.