I have a question relevant to the number of graphs with prescribed spectral ratio. Let $A$ be the adjacency matrix of a graph on $n$ vertices. Let $\lambda_i$ be its $i$-th largest (signed) eigenvalue. Moreover, assume the following (larger than a constant) spectral ratio constraint $$\frac{\lambda_1}{\max\{\lambda_2,|\lambda_n|\}} \ge \Omega(\log(n)).$$ What is the fraction of graphs (out of the $2^{O(n^2)}$ many) that have the above spectral property?
In other words, what is the fraction of graphs on $n$ vertices with increasing spectral ratio?
Edit: A relevant question that might be easier: 'is there a combinatorial property on the graph that leads to an $Ω(\log(n))$ ratio?'