This work studies super-Ramanujan graphs: \(d\)-regular graphs whose spectral expansion lies strictly below the Ramanujan threshold \(2\sqrt{d-1}\). Although this threshold is asymptotically optimal for fixed \(d\) as \(n\) tends to infinity, it leaves open the finer question of how small the spectral expansion can be as a joint function of \(n\) and \(d\). Beyond their fundamental mathematical interest and potential applications to pseudorandomness, super-Ramanujan graphs may also have practical significance, since asymptotic analyses can obscure important finite-size effects.

Our first result shows that, for every \(d\geq 3\) and every even \(n\), there exists a \(d\)-regular graph \(G\) on \(n\) vertices satisfying
\[
\lambda_2(G)
\leq
2\sqrt{d-1}
-
\frac{\sqrt{d}}{4n^{2/3}}.
\]
For each fixed \(d\), an asymptotic version of this bound also follows from the recent breakthrough work of Huang, McKenzie, and Yau. Related edge-universality results of Huang and Yau (The Annals of Probability, 2026) imply a bound of the same order in the regime \(
n^\varepsilon\leq d\leq n^{1/3-\varepsilon},
\) for any fixed sufficiently small \(\varepsilon>0\), while He (Commun. Math. Phys., 2024) obtains a substantially larger advantage in the denser regime \(d\gg n^{2/3}\). These works provide a much richer probabilistic description of the spectral edge, whereas our proof is considerably simpler and gives a nonasymptotic bound that is uniform in both \(n\) and \(d\).

Turning to explicit constructions, it is folklore that the Lubotzky-Phillips-Sarnak graphs are super-Ramanujan, but their guaranteed advantage over the Ramanujan threshold is only exponentially small in \(n\). Our second result gives a polynomial-time construction of bipartite super-Ramanujan expanders with advantage at least \(
\frac{\sqrt{d}}{2n}
\) over the Ramanujan threshold.