Reflexive (hyperbolic) graphs 
Is there an effective description of the graphs such that exactly one eigenvalue (of the conventional adjacency matrix) is $>2$ whereas all others are $\le2$?

By "effective" I mean something that would let one iterate through all such graphs with reasonably few (say, a few dozens) vertices, or, better yet, to prove theorems about such graphs :) (Ideally, a list similar to that of Dynkin diagrams.)
The question is motivated by this one: "spectrum of an adjacency matrix" and by my own research. It seems wild, but I'm not an expert.
 A: I think this question is so hard, since we do not have any control on other eigenvalues, specially on the minimum of them. 
As an evidence (and maybe useful for your work), recently S. M. Cioab‎$‎\breve{a}‎$‎, W. H. Haemers, J. Vermette and W. Wong‎ in the paper with name:
"The graphs with all but two eigenvalues equal to ‎‎$‎\mp‎$‎‎ 1",
characterized all graphs that have two eigenvalues $r>1$ and $s<-1$, and all other eigenvalues are $1$ and $-1$. The way of their proof is very interesting and special.
So, if $r>2$ and all other eigenvalues is less than or equal $1$, we have very difficult task to describe these graphs.
Do you have any evidences that hopeful you for solving this problem in general?
A: I do not recall seeing such a characterization.
However Neumaier has looked at some related stuff. In A. Neumaier, J. J. Seidel "Discrete hyperbolic geometry" they consider graphs where the second largest eigenvalue is equal to 2. The paper Renee Woo, Arnold Neumaier "On Graphs Whose Spectral Radius is Bounded by 3/\sqrt{2}" might also contain something of interest.
