Graphical Partitions

[Gordon Royle]

On 31/05/2006, at 4:22 AM, Eric W. Weisstein wrote:

> On Tue, 30 May 2006, franktaw at netscape.net wrote:
>
>> Sequences A000569 and A004250 are both described as "Number of  
>> graphical partitions of 2n."  However, they are different.
>> Likewise, from their descriptions A004251 and A095268 both appear  
>> to be the number of distinct graphical partitions for a graph with  
>> n nodes; but again they are different.
>
> http://www.research.att.com/~njas/sequences/A004250 is the number  
> of n-node graphical partitions (=simple graphs with no isolated  
> points):
>

I am not clear at all why the number of graphical partitions should  
be the number of graphs with no isolated points with the usual  
definitions of equivalence for both objects.

For example, there are two non-isomorphic graphs on 5 vertices with  
the degree sequence {1,1,1,2,2}.

These two graphs contribute "2" to the "number of graphs with no  
isolated points"

However, we would normally say that they contribute just "1" to the  
number of graphical partitions, because they both yield the same  
partition.


Basically it would seem that you are DEFINING "graphical partition"  
to mean the PAIR  ("graph", "degree sequence") such that two  
graphical partitions are defined to be equal if and only if the  
"graph" co-ordinates are isomorphic.


Of course you are free to define anything in any way you choose, but  
I would suggest that this is sufficiently unusual that it warrants a  
special warning to the reader...

Gordon