It is known that there is a gap between 2 and the next largest norm of a graph. Is there an interval of the real line in which norms of graphs are dense?

I found a reference that seems to answer your question: Shearer, James B. On the distribution of the maximum eigenvalue of graphs, 1989. The theorem in this paper is that the set of largest eigenvalues of adjacency matrices of graphs is dense in the interval $\left[\sqrt{2+\sqrt{5}},\infty\right)$. Here's an online version. Here's a related paper: Hoffman, Alan J. On limit points of spectral radii of nonnegative symmetric integral matrices, 1972. In this paper limit points less than $\sqrt{2+\sqrt{5}}$ are described. In particular, they form an increasing sequence starting at 2 and converging to $\sqrt{2+\sqrt{5}}$. Here's an online version. The author also posed the problem that led to Shearer's paper. 

