[seqfan] Re: A064513, A058201, and the graph degree/diameter problem.

[Allan Wechsler]

a(8) <= 65 by the Moore bound. Since 8 is not in {2,3,7,57}, we know a(8)
<= 64. I don't know if we have any better upper bounds. This seems like a
decent undergraduate research project. Pushing up the lower bound also.

On Wed, Nov 18, 2015 at 10:12 AM, Chris Thompson <cet1 at cam.ac.uk> wrote:

> On Nov 17 2015, Neil Sloane wrote:
>
> Allan, Chris,
>> Thanks for those comments.
>> I "deaded" A058201, and I'm revising A064513.
>>
>> My old notes there say that it is known that a(7)=50. Do you believe that?
>>
>
> That has been known since the first construction of the Hoffman-Singleton
> graph: probably http://dx.doi.org/10.1147/rd.45.0497
>
> The Moore upper bound, a(n) <= n^2 + 1 in this case, can be achieved with
> equality only for n = 2, 3, 7 or 57. The first three cases give unique
> graphs (5-cycle, Petersen graph, Hoffman-Singleton graph) while the
> existence
> or otherwise of a Moore graph with degree 57 is a notorious open problem.
>
> Apart from the Wikipedia page that Allan mentioned, there is one for
> the Hoffman-Singleton graph
>
>  https://en.wikipedia.org/wiki/Hoffman%E2%80%93Singleton_graph
>
> with plenty more references.
>
>
> --
> Chris Thompson
> Email: cet1 at cam.ac.uk
>
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