In the early 80's McKay proved that, as the number of vertices goes to infinity, the eigenvalue distribution of X^N, the adjacency matrix of a random d-regular graph, converges to the now called Kesten-McKay distribution. He then used this to determine the asymptotic growth of the expected number of spanning trees in a random d-regular graph. 

In this work, we extend the polynomial method (introduced in [Chen, Garza-Vargas, Tropp, van Handel 24]) to show that for any smooth function h that is centered with respect to the Kesten-Mckay distribution, the linear spectral statistic Tr h(X^N) converges to a Poisson mixture. We then use this to derive the asymptotic distribution of the fluctuations of the number of spanning trees in a random d-regular graph. 

This is upcoming work with Ramon van Handel.