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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?

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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.

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