The uniqueness of a Moore graph of degree 57 and diameter 2 is not known. See, for instance, this paper -
- where they refer to `the missing Moore graph(s)' to indicate this fact.
Other discussion can be found here:
Edit: I have browsed the paper of Macaj and Siran linked to above, the main result of which says that if a Moore(57,2)-graph exists then its automorphism group has order at most 375. Let me give a relevant quote from the paper:
On the other hand, in the study of
possible actions of groups of order
375 with 10 orbits we found hundreds
of matrices satisfying conditions of
Lemma 5 which we were not able to
exclude by our techniques.
(The `matrices satisfying conditions of Lemma 5' are adjacency matrices for particular partitions of a putative Moore(57,2)-graph.) In other words, in theory there could be many different Moore(57,2)-graphs even if you just restrict to those with an automorphism group of order 375.