About some possible optimality properties of Ramanujan graphs
At least among regular bi-partite graphs, Ramanujan graphs provide the optimal approximation of the complete bipartite graph. Let's say that a graph $H$ $C$-approximates a graph $G$ if $tL_H \preceq CL_G$ for $t$ the smallest real number such that $L_G \preceq tL_H$. (I.e. $t := \|L_H^{-1/2} L_G L_H^{-1/2}\|$ where the norm is the operator norm and inverses are taken w.r.t. to the space orthogonal to the all-ones vector). The eigenvalues of $L_{K_{n,n}}$ are $\lambda_1 = 0, \lambda_2 = \ldots \lambda_{2n-1} = n, \lambda_{2n} = 2n$. For any $d$-regular bipartite $H$, the eigenvalues $0=\mu_1 \le \ldots \le \mu_{2n} = 2d$  of $L_H$ come in pairs $d \pm x$. By Alon-Boppana, $\mu_2 \le d-2\sqrt{d-1}$, and, therefore, $\mu_{2d-1} \geq d + 2\sqrt{d-1}$ (ignoring lower-order terms). This implies $C \geq \frac{d + 2\sqrt{d-1}}{d - 2\sqrt{d-1}}$, which is achived by a $d$-regular Ramanujan graph.