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.

onepositive square and any number of zeroes and/or negative squares. $\endgroup$