In this Letter we propose a simple algebraic recursion for the complete one-loop integrands of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>N</a:mi></a:math>-graviton correlators. This formula automatically yields the correct symmetry factors of individual diagrams, taking into account both the graviton and the ghost loop, and seamlessly controlling the related combinatorics. Published by the American Physical Society 2025