Is there a continuous analogue of Ramanujan graphs? I think it might help to think of the following definition of a Ramanujan graph - a graph whose non-trivial eigenvalues are such that their magnitude is bounded above by the spectral radius of its universal cover. 
By "non-trivial eigenvalues" I mean all the eigenvalues except the highest and the smallest. A universal cover of a graph is the infinite tree such that every connected lift of the graph is a quotient of the tree. The spectral radius of a graph would be the norm of its adjacency matrix. 
It would be helpful if people can give any pointers along these directions..
 A: One possible answer can be found in the theory of graph limits, where large graphs are modelled by continuous objects.  In particular, a graphing is one type of continuous analogue of a graph, and it still makes sense to do spectral theory on graphings.  Thus, one can define a suitable notion of a Ramanujan graphing.  See Section 2 of this paper of Backhausz, Szegedy and Virág, where this is carried out.  
A: In fact, the original motivation behind Lubotzky--Phillips--Sarnak's construction of Ramanujan graphs was in analogy with modular curves $Y(N)=\mathbb H^2/\Gamma(N)$ for the principal congruence subgroups $\Gamma(N)\subseteq\operatorname{PSL}(2,\mathbb Z)$.  So the answer is yes, there is a continuous analogue, but in fact it came first!
Let me give a few more details.  The spectrum of the Laplacian $\Delta$ on hyperbolic space $\mathbb H^2$ consists of $[\frac 14,\infty)$.  Selberg proved that the smallest positive eigenvalue of the Laplacian on $Y(N)$ satisfies $\lambda_1(Y(N))\geq\frac 3{16}$, and conjectured that $\lambda_1(Y(N))\geq\frac 14$.  Note that $\frac 14$ is exactly the inf of the spectrum of $\Delta$ on the universal cover $\mathbb H^2$.  You can read more about this in an article by Sarnak.
As far as I understand things, Lubotzky--Phillips--Sarnak's examples of Ramanujan graphs are discrete analogues of modular curves.
A: Check this out
http://arxiv.org/abs/1301.1028
It's Ramanujan complex
