Central limit theorems for linear spectral statistics of inhomogeneous random graphs with graphon limits

We establish central limit theorems (CLTs) for the linear spectral statistics of the adjacency matrix of inhomogeneous random graphs across all sparsity regimes, providing explicit covariance formulas under the assumption that the variance profile of the random graphs converges to a graphon limit. Two types of CLTs are derived for the (non-centered) adjacency matrix and the centered adjacency matrix, with different scaling factors when the sparsity parameter $p$ satisfies $np=n^{\mathrm{\Omega }(1)}$, and with the same scaling factor when $np=n^{o(1)}$. In both cases, the limiting covariance is expressed in terms of homomorphism densities from certain types of finite graphs to a graphon. These results highlight a phase transition in the centering effect for global eigenvalue fluctuations. For the non-centered adjacency matrix, we also identify new phase transitions for the CLTs in the sparse regime when $n^{1\mathrm{/}m}\ll np\ll n^{1\mathrm{/}(m-1)}$ for $m\ge 2$. Furthermore, weaker conditions for the graphon convergence of the variance profile are sufficient as $p$ decreases from being constant to $np\to c\in (0,\mathrm{\infty })$. These findings reveal a novel connection between graphon limits and linear spectral statistics in random matrix theory.