From: ptwahl@aol.com (PTWahl)
Newsgroups: sci.math
Subject: Re: Biggest Moore Graph
Date: 1 Dec 1998 19:14:41 GMT

On Mon, 30 Nov 1998 18:15:09 -0000, "mongrels"
<mongrels@streamlinerec. nospamprestel.co.uk>
wrote:

> Subject: Biggest Moore Graph

>Has anyone found this or disproved its existence yet?
>
>Kesh McMongrel

You may be referring to the "Moore graph of diameter
= 2, with (all) vertices of degree = 57" ... as far as I know,
there is some evidence to restrict this graph, but it *might*
exist.  It cannot have a property, which I do not want to define here, that the
smaller ones (pentagon, Petersen,
Hoffmann-Singleton) of diameter = 2 all have.  That does
not prove nonexistence, but we know  it "can't be pretty."

H and S proved (1960, I think) that the H-S graph exists
at degree = 7, and for diameter = 2 that degree = 57 is
the only remaining possibility.  Biggs (1994) in _Algebraic
Graph Theory_  explains the restriction I mentioned, and
gives references to the published papers.

I only read about this from time to time, and am not an
expert.  However, web sites that should be up to date
have not mentioned any breakthrough.  This would be
high on my list of "neat mathematical objects" if it is
ever found.

Regards,
Patrick T. Wahl
Greeley, Colorado, USA