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April 7th, 2017, 06:48 AM   #1
ZMD
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Unhappy Adjacency matrix

Consider the graph with three vertices A,B, C such that one can go from any vertex to any other vertex in one step.

1.Compute the adjacency matrix L of this graph.

2.Let $a_{n}$ denote the number of paths of length n that start at A and end at A.

For instance, since A $>$ B $>$ A and A $>$ C $>$ A are the only such paths of length 2,we have $a_{2}$ =2.By computing the powers $L^{n}$ show that:
$a_{n}$ =1/3 * ($2^{n}$ + 2$(-1)^{n}$)
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April 11th, 2017, 03:43 AM   #2
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If "one can go from any vertex to any other vertex in one step" then the adjacency matrix is simply . It should be easy to see that the nth power of L is just the matrix with in every entry.
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