Re: Unlabeled graphs

[Charles Wells <charles@freude.com>]

This is in reply to Eduardo Dubuc (quoted after the reply).  I have a major
point and a minor point.

Minor point: The phrase "labeled graph" usually allows different nodes to have
the same label.  Eduardo clearly intends, however, that the labeling be
injective.  (In the usual sense there are n^n labels, not n!).  This is only a
matter of terminology.  My answer refers to injective labeling as he intended. 
(Better terminology for his purposes would be to call them NAMED nodes.)

Major point:  Any way you store an unlabeled graph in a computer will involve a
memory location for each node, and so introduces a labeling, indeed a total
ordering, on the nodes.  However, a high level language with pointers such as C
can be used to describe the structure without any names being given
explicitly.  The structure head will point to a node, which will have a pointer
to another node, etc, and the program text will not mention all the nodes
directly.  For the purpose of computing with the graph, you can have the
program traverse the nodes without naming them.  This is a commonly used
technique.  (There will have to be pointer structures for the edges with
pointers to the two nodes each is incident on.)  When the program is compiled,
the nodes are made to correspond to memory locations but the locations are
hidden from the user.  It seems to me that this preserves the psychology of a
drawing with no labels on the nodes.

You can respond that each node has an implicit label, essentially an integer,
which is the number of pointers you have to dereference to get to the node. 
But that is just like saying that in the drawing of the graph on a piece of
paper, each node has an implicit label, namely its location on the paper.  If
you allow implicit labels like this the drawing has them and so does the graph
in the computer.  If you don't allow them then neither has labels.

Let me forestall someone bringing up a red herring:  The coordinates of the
node on the paper depend on an arbitrary choice of coordinate system, unlike
the number of pointers you need to get to the stored node in the computer. 
This argument, if someone makes it, is based on a misreading of what I said.  I
said each node is labeled by its location on the paper.  I didn't say the
coordinates of the location.  The location itself is the label.  (But how do
you talk about it?  You POINT to it!)

--Charles Wells


>by an unlabeled graph i mean the drawing of a graph, in paper, say, or a
>graph buildt in space, the skeleton of a building for example. It has
>vertices and edges, and everybody knows what it is. Mathematically you
>could say a symetric relation on its (finete) set of vertices. But not
>quite so ...
>
>If you have n vertices, you have n! different labeling. Each labeling
>gives you a different labeled graph.
>
>The minute you have a set (in the mathematical sense) of vertices, you
>have a labeling. Namely, the elements of that set are the labels!. So,
>with a symetric relation (in the mathematical sense) what you got is a
>labeled  graph. Not an unlabeled graph !.
>
>And you become well aware of this fact when you want to put a concrete
>unlabeled graph (say, the skeleton of a  building) inside a computer !!
>
>REMEMBER I rise the question on unlabeled graph related to the question
>that we were discussing:
>
>INEVITABILITY OF NAMING (IN MATHEMATICS) 
> (naming is not the same as labeling ?)  
>
>>> well, unlabeled graph has to be a quotient by an equivalent relation 
>...
>
>I said that. It seems possible. I explain now the  ...
>
>  Given two graphs R, S (symetric relations) on a finite set X (of 
>vertices), consider the natural action of the symetric group of X on the
>power set of X x X. Then, R =~ S iff they are in the same orbit. The
>elements of the quotient set are the unlabeled graphs.
>
>    e.d.
>
>
>
>
>
>



Charles Wells, 105 South Cedar St., Oberlin, Ohio 44074, USA.
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