Random graphs, expanding families and the construction of noncompact hyperbolic surfaces with uniform spectral gaps

In this paper, we introduce and analyze a random graph model ${\mathcal{F}}_{\chi ,n}$, which is a configuration model consisting of interior and boundary vertices. We investigate the asymptotic behavior of eigenvalues for graphs in ${\mathcal{F}}_{\chi ,n}$ under various growth regimes of $\chi$ and $n$. When $n=o\left({\chi }^{\frac{2}{3}}\right)$, we prove that almost every graph in the model is connected and forms an expander family. We also establish upper bounds for the first Steklov eigenvalue, identifying scenarios in which expanders cannot be constructed. Furthermore, we explicitly construct an expanding family in the critical regime $n\asymp g$, and apply it to build a sequence of complete, noncompact hyperbolic surfaces with uniformly positive spectral gaps.