Researchers Christian Hirsch, Kyeongsik Nam, and Moritz Otto have established a central limit theorem for linear eigenvalue statistics within random geometric graphs. These networks, where connections depend on geometric proximity, increasingly model systems constrained by spatial structure, yet their spectral properties have remained poorly understood compared to classical random graph models. This work presents the first rigorous analysis of Gaussian fluctuations for linear eigenvalue statistics, demonstrating a central limit theorem for a broad class of test functions and, in the polynomial case, providing a quantitative convergence rate. By extending these findings to other canonical random spatial networks, including k-nearest neighbour graphs and relative…