Critical small-set edge expansion threshold in bipartite random regular graphs

Ascertain the computational hardness of certifying the exact critical bound \(\Phi_{\varepsilon}^{e}(G) \ge d - 1 - \sqrt{d-1}\) (with no error terms) for bipartite random d-regular graphs \(G \sim \mathcal{G}((n/2,n/2), d)\) when \(d=q^{2}+1\) (q a prime power) in the limit \(\varepsilon \to 0\). Determine whether polynomial-time certification at this threshold is possible or provably hard.

Background

Analogously to vertex expansion, the authors use explicit bipartite Ramanujan graphs to argue (conditionally) that Kahale’s spectral bound is optimal up to leading order for small-set edge expansion.

They explicitly point out the unresolved status exactly at the critical edge expansion threshold for certain degrees $d=q^{2}+1$, leaving open whether polynomial-time certification is achievable or hard at this precise value.