Spectral bounds for superspecial abelian-surface isogeny graphs

Let $p\ge 5$ and let $\ell\ne p$ be prime. Let $\mathcal{G}^{SS}_{2}(\ell,p)$ be the superspecial isogeny graph, with normalized adjacency eigenvalues

$$1=\mu_1>\mu_2\ge\cdots\ge\mu_m>-1,$$

where $m=|SS_2(p)|$, and set $\lambda_i=1-\mu_i$. Write $N_2(\ell)$ for the graph degree. Spectral-bound conjecture. For every $i$,

$$1-\frac{\max\{4\ell\sqrt{\ell},\ell^2+1+2\ell\sqrt{\ell}\}}{N_2(\ell)}\le \lambda_i\le 1+\frac{4\ell\sqrt{\ell}}{N_2(\ell)}.$$

In particular, for $\ell=2$,

$$1-\frac{8\sqrt{2}}{15}=0.24575\ldots\le\lambda_2,\qquad 2-\lambda_m\le 1+\frac{8\sqrt{2}}{15}=1.75425\ldots.$$

The proposed bounds combine the expected estimate for non-CAP forms with the larger Saito–Kurokawa contribution; the source gives no resolution of this spectral statement.