In *Discrete groups, expanding graphs, and invariant measures* (in the notes at the end of Chapter 7), Lubotzky mentions that certain estimates for the number of spanning trees $\kappa(G)$ of a $k$-regular graph $G$ have nontrivial arithmetic consequences such as estimates for the class number of certain interesting function fields, citing an unpublished paper of Sarnak from 1988, *Number theoretic graphs*, which I can't find anywhere on the internet.

I have heard that the class number for the function field of the modular curve $X(N)$ over $\mathbb{F}_p$ is related to the number of spanning trees in the $(p+1)$-regular supersingular isogeny graph $G(N,p)$, whose vertices are supersingular elliptic curves over $\mathbb{F}_N$ and edges are degree $p$ isogenies. I'm not sure about the details of this, so please correct me if I'm wrong here.

Is this the connection Lubotzky was talking about? If so I must be misunderstanding something because Lubotzky mentions that the combinatorial result which has arithmetic corollaries is the following lower bound due to Alon:
$$\beta(k)=k-O\left(k \frac{(\log \log k)^2}{\log k}\right)$$
where $\beta(k)$ is the limit inferior of $\kappa(G)^\frac{1}{|G|}$ over all $k$-regular graphs. On the other hand, I believe that, for supersingular isogeny graphs, as $N$ goes to infinity the girth also goes to infinity (correct me if I'm wrong), and for graphs with girth going to infinity it is known that $\kappa(G)^\frac{1}{|G|}$ approaches the constant $\frac{(k-1)^{k-1}}{(k(k-2))^{\frac{k}{2}-1}}$ (see e.g. *Lyons, Russell*, **Asymptotic enumeration of spanning trees**), so Alon's result doesn't tell us anything in this case.

Question 1:Does anyone know where I can find this paper of Sarnak, to help sort this out?

Question 2:Am I right that Alon's estimate doesn't say anything about supersingular isogeny graphs, and if so what is the application of Alon's estimate that Lubotzky had in mind?

**Edit:** I believe that my claim that the girth of supersingular isogeny graphs go to infinity with $N$ is actually false (I confused with other related graphs). On the other hand, I think it is still true that these graphs converge in the Benjamini-Schramm sense to the $k$-regular tree, so we still have that $\kappa(G)^\frac{1}{|G|}$ approaches $\frac{(k-1)^{k-1}}{(k(k-2))^{\frac{k}{2}-1}}$.