This is the [Moore graph][1], which is a regular graph of degree $d$ with diameter $k$, with maximum possible nodes. A calculation shows that the number of nodes $n$ is at most

$$
1+d \sum_{i=0}^{k-1} (d-1)^i
$$

and as you mentioned it can be shown by spectral techniques that the only possible values for $d$ are

$$ d = 2,3,7,57. $$

Example for $d=7$ is the [Hoffman–Singleton graph][2], but for the case $d=57$ it is still open. See Theorem 8.1.5 in the book "[Spectra of graphs][3]" by Brouwer and Haemers for reference.


  [1]: http://en.wikipedia.org/wiki/Moore_graph
  [2]: http://en.wikipedia.org/wiki/Hoffman-Singleton_graph
  [3]: http://homepages.cwi.nl/~aeb/math/ipm.pdf