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 April 7th, 2017, 06:48 AM #1 Member   Joined: Nov 2016 From: Kansas Posts: 73 Thanks: 1 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$(-1)^{n}$)
 April 11th, 2017, 03:43 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,197 Thanks: 872 If "one can go from any vertex to any other vertex in one step" then the adjacency matrix is simply $\begin{pmatrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}$. It should be easy to see that the nth power of L is just the matrix with $3^n$ in every entry.

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