[seqfan] Re: Multiplicative Graphs
Andrew Weimholt
andrew at weimholt.com
Fri Oct 23 03:06:18 CEST 2009
I am also tempted to conjecture that the only odd terms are a(p)=1, for prime p
Andrew
On 10/22/09, Andrew Weimholt <andrew at weimholt.com> wrote:
> Hi Franklin,
>
> nice idea for a sequence!
> Your sequence is missing either a(9) or a(10)
>
> > 1,1,1,1,2,1,2,1,4,2,1,6
>
> should be
> 1,1,1,1,2,1,2,1,4,2,2,1,6,...
>
> Also, by hand, I get a(36) = 24, not 23.
>
> and I get a(48) = 42
>
>
> Andrew
>
>
>
> On 10/22/09, franktaw at netscape.net <franktaw at netscape.net> wrote:
> > For a given graph, suppose we take the product of the valences of the
> > vertices.
> >
> > The question is how many graphs have a given product. To avoid
> > infinities, we will consider only one-component graphs; that is,
> > connected graphs excluding the empty graph. (If we allow the the empty
> > graph, a(1) will be one larger.)
> >
> > Since valence 1 nodes don't directly affect the product, we can look at
> > the graph that results when they are removed. This will be a connected
> > graph, and labeling each node with its original valence, the product of
> > the labels will be the original product. Each node must be labeled
> > with at least its valence, and at least 2. Furthermore, each such
> > labeling, up to graph equivalence, uniquely defines the original graph
> > -- so this gives us a way to compute the sequence.
> >
> > For a given n, we need only look at connected graphs with at most
> > BigOmega(n) (A001222) nodes. In particular, if n is prime, a(n) = 1,
> > and if n is a semiprime, a(n) = 2.
> >
> > Hand-calculating, and starting with n = 0 (the one-point graph), I get:
> >
> > 1,1,1,1,2,1,2,1,4,2,1,6,1,2,2,8,1,6,1,6,2,2,1,16,
> > 2,2,4,6,1,8,1,16,2,2,2,23,1,2,2,16,1,6,1,6,6,2,1
> >
> > I don't know what a(48) is.
> >
> > I would appreciate it if someone could (1) verify these values, and (2)
> > compute some more. Anything else anyone can contribute about the
> > sequence would also be welcome.
> >
> > Franklin T. Adams-Watters
> >
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
> >
>
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