# Are the norms of graphs dense in any interval?

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?

• What is the norm of a graph? May 9 '10 at 4:56
• The norm of a graph is the largest eigenvalue of its adjacency matrix. Scott's question mathoverflow.net/questions/1822/… has some motivation for thinking about graph norms. May 9 '10 at 5:35

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.
Hoffman, Alan J. On limit points of spectral radii of non-negative 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.
• The set of limit points was vaguely reminiscent of the set of possible Jones indices, which is why I looked. I found the proposition using the listing of $(2+5^{1/2})^{1/2}$ in the index. I then hesitated to mention it, not wanting the comment to be seen as criticism--but I had to. We haven't met, but I imagined it would be worth a laugh. No need to apologize. May 9 '10 at 17:34