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Complete Bipartite Graphs

  1. Apr 21, 2012 #1
    1. The problem statement, all variables and given/known data
    How many complete bipartite graphs have n vertices?


    2. Relevant equations



    3. The attempt at a solution
    I said, let the first bipartition set have x vertices and the second bipartition set have y vertices. So x+y=n.

    I think that you would do C(n, x)*C(n-x, y)=C(n,x)*C(y,y)=C(n,x) but I feel like this is either wrong or not good since I made x and y up arbitrarily.

    Help?
     
  2. jcsd
  3. Apr 21, 2012 #2

    Office_Shredder

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    So that's the number of complete bipartite graphs with the "first" partition having size x. What are the possible values for x?
     
  4. Apr 21, 2012 #3
    There are n possible values for x... so is the answer n*C(n,x)?

    Is there any problem with me having made x up without it being in the problem?

    Also, would I instead want to do something like C(x,1)*C(y,1) [to get the number of ways to pick the vertices] *C(n,x) [to get the number of ways of making the 'first' bipartition set] *n?
     
  5. Apr 22, 2012 #4
    Could it matter if there is an even/odd number of vertices in any one bipartition set?
     
  6. Apr 22, 2012 #5
    Any new ideas?
     
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