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 Post subject: Adjacency matrixPosted: Sun, 10 Apr 2011 11:24:17 UTC
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Joined: Sat, 26 Mar 2011 16:45:36 UTC
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There is a connected graph on 15 vertices. Let be A the matrix that we get from its adjacency matrix by writing ones to its main diagonal (the rest of the elements remain unchanged). Prove that all of the elements of are positive integers.

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 Post subject: Re: Adjacency matrixPosted: Sun, 10 Apr 2011 12:14:15 UTC
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mrgrieco wrote:
There is a connected graph on 15 vertices. Let be A the matrix that we get from its adjacency matrix by writing ones to its main diagonal (the rest of the elements remain unchanged). Prove that all of the elements of are positive integers.

Silly problem. Find a path and stays there.

More interesting would be to impose and not add the loops.

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 Post subject: Re: Adjacency matrixPosted: Sun, 10 Apr 2011 14:44:53 UTC
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Joined: Sat, 26 Mar 2011 16:45:36 UTC
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outermeasure wrote:

Silly problem. Find a path and stays there.

More interesting would be to impose and not add the loops.

To tell the truth, I didn't really catch the point. Could you please explain it?
Thank you!

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 Post subject: Posted: Sun, 10 Apr 2011 18:14:39 UTC
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Each entry (i,j) of A^20 is nonzero iff there is a path of length 20 between node i and node j. Because the graph is connected, there is always a path between any two nodes. Because of the 1s on the diagonal, each node has a self-adjoining edge, and therefore it is trivially possible to increase the length of any non-cyclic path to 20.

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