I found a reference that seems to answer your question:

Shearer, James B.
[On the distribution of the maximum eigenvalue of graphs][1], 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][2].


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Here's a related paper:

Hoffman, Alan J.
[On limit points of spectral radii of non-negative symmetric integral matrices][3], 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][4].    The author also posed the problem that led to Shearer's paper.


  [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=986863
  [2]: http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V0R-45W3BTD-1V&_user=440026&_coverDate=04%252F30%252F1989&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000020939&_version=1&_urlVersion=0&_userid=440026&md5=99474a7c10cf69e8e86314cf27d6507a
  [3]: https://mathscinet.ams.org/mathscinet-getitem?mr=347860
  [4]: https://doi.org/10.1007/BFb0067367