This is the Moore graph, 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, but for the case $d=57$ it is still open. See Theorem 8.1.5 in the book "Spectra of graphs" by Brouwer and Haemers for reference.