n choose k problem with graph structure

Suppose we have a square (m x m), which consists of m^2 empty boxes.
If we want to put k cards to the square, we have a typical n-choose-k
problem: the total number of patterns is m^2-choose-k.

Now, if we consider each pattern is a graph with each card as a
node(the distance between two nodes is a edge), we can see many
identical patterns among the initial m^2-choose-k. We need to discount
those idential patterns and count the total, what will it be?

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Clarification of Question by leiye-ga on 25 Dec 2005 12:17 PST

Hi, Thanks for your comments. Let make it more clear. suppose m=3,
k=3. The following three patterns will be counted different in
m^2-choose-3, right?

Pattern a):  'x' means a node there, '0' means empty
X 0 0
0 X 0 
X 0 0

Pattern b):
0 X 0
0 0 X 
0 X 0

Pattern c):
X 0 X
0 X 0 
0 0 0

But if we consider they are a graph (x is the node) and no location is
considered, then b) will be same a).  If c) is rotated, then c) will
be same as a) and b).

Therefore, my question actually is: in a m x m area, how many m-node graphs?

I'm good enough at math to understand any question as complex as the
one you are trying to ask -- that said, I do not know what you are
asking. I don't think anyone else will know either.

Show an example of two configurations that are "identical" and should
not be double-counted, and it might be possible for someone to answer
your question.

Also this is not a "computers" question. I think that a math question
would go under "Science".

Hi, Thanks for your comments. Let make it more clear. suppose m=3,
k=3. The following three patterns will be counted different in
m^2-choose-3, right?

Pattern a):  'x' means a node there, '0' means empty
X 0 0
0 X 0 
X 0 0

Pattern b):
0 X 0
0 0 X 
0 X 0

Pattern c):
X 0 X
0 X 0 
0 0 0

But if we consider they are a graph (x is the node) and no location is
considered, then b) will be same a).  If c) is rotated, then c) will
be same as a) and b).

Therefore, my question actually is: in a m x m area, how many m-node graphs?

You did not define the "edge" in the graph.