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Math Help - How to calculate cycles of a length using an adjancy matrix?

  1. #1
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    How to calculate cycles of a length using an adjancy matrix?

    I have an adjancy matrix, and I have to calculate the number of cycles of length 3 that is paths of length 3 that have the same start and end matrix: this is the matrix:
    (0 1 1)
    (1 1 1)
    (0 1 0)

    This is a past paper without answers and would really help me revise, thanks in advance !
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  2. #2
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    Look at the powers of that matrix. Do you see that the square shows where you can go in two steps? What about the cube?
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  3. #3
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    So if I take the cube of the matrix I get:
    ( 2 4 3 )
    ( 3 6 4 )
    ( 1 3 2 )
    If I add these will I get the amount of cycles of length 3 ?
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  4. #4
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    If A is an adjacency matrix, then the (i, j)th component of A^n is the number of paths of length n from i to j (see MathWorld). You need loops, so consider the diagonal elements.
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