April 7th, 2017, 06:48 AM  #1 
Member Joined: Nov 2016 From: Kansas Posts: 70 Thanks: 0  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: 3,092 Thanks: 845 
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.


Tags 
adjacency, matrix 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
ultra diagonal matrix, super diagonal matrix, sub diagonal matrix  silviatodorof  Linear Algebra  2  March 22nd, 2015 05:28 AM 
H matrix?  dervast  Linear Algebra  1  March 2nd, 2011 12:32 AM 
matrix  ely_en  Algebra  3  March 8th, 2010 10:38 AM 