# 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..

• It would be helpful if the question were not merely in the title, but asked and expanded upon in the body of the question. It's hard figuring out what you want. – Todd Trimble Nov 5 '14 at 12:27

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.

• Pradon Thanks for the explanations. But the link is not opening. Can you kindly link to some pedagogical article on this ? Something publicly available? – Anirbit Nov 7 '14 at 14:04

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.

Check this out http://arxiv.org/abs/1301.1028 It's Ramanujan complex

• While the paper is interesting and relevant, as far as I can see, Ramanujan complexes are finite, so it's a generalization in a different direction than what the answer asks for. – Emil Jeřábek Nov 5 '14 at 12:42
• @EmilJeřábek: perhaps you rather mean "... than what the question asks for"? – Stefan Kohl Nov 5 '14 at 13:27
• if the author means a generalization to simplicial complexes then I guess there is not yet an agreement on the "right definition" of expanders (which Ramanujan graphs are extreme cases) – Son P Nguyen Nov 5 '14 at 15:44
• I believe the issue is that the author asks in the title for continuous analogues, not higher-dimensional analogues. – j.c. Nov 5 '14 at 16:25