In the related literature one often sees the phrase "The missing Moore graph" which (to me) tacitly implies that the missing Moore graph (if exists) is unique.


Is there a result of this type or is or is this just a limitation of words that do not express the fact that there could be nonisomorphic graphs of diameter 2 and degree 57?



(The) missing Moore graph(s) - uniqueness
The uniqueness of a Moore graph of degree 57 and diameter 2 is not known. See, for instance, this paper -
<http://www.sciencedirect.com/science/article/pii/S0024379509003735>
- where they refer to `the missing Moore graph(s)' to indicate this fact.


Other discussion can be found here:
<http://symomega.wordpress.com/2009/09/11/i-want-more-moore-graphs/>


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.


