hw4soln

hw4soln - CME 305: Discrete Mathematics and Algorithms...

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CME 305: Discrete Mathematics and Algorithms Instructor: Professor Amin Saberi (saberi@stanford.edu) HW#4 – Due 03/18/11 1. In this problem, we use the reﬂection principle from the study of partial diﬀerential equations to ﬁnd the eigenvalues and eigenvectors of the path graph, i.e. two vertices have degree one and all other vertices have degree two. (a) Write the Laplacian matrix for the path graph on N vertices, and for the ring graph (a simple cycle) on 2 N vertices. (b) Show that if v R N is an eigenvector for the path graph, then w = ( v 1 ,v 2 ,...,v n ,v n ,v n - 1 ,...,v 1 ) > is an eigenvector for the ring graph on 2 N vertices. Likewise, show that if w is an eigenvector of the ring graph obeying w i = w 2 N +1 - i for i = 1 ,...,N , then v = ( w 1 ,...,w n ) > is an eigenvector for the path graph. (c) Recall the geometric argument from class showing that the eigenvectors of the ring graph on N vertices are ( x k ) i = cos ± 2 πki N ² and ( y k ) i = sin ± 2 πki N ² Use this to ﬁnd the eigenvalues and eigenvectors of the path graph on N vertices. Hint: Both x k and y k have the same eigenvalue. So any vector in the space spanned by these vectors is an eigenvector with the same eigenvalue. (d) This trick is called the reﬂection principle. If a neighbor in a mirror is treated exactly as a real neighbor, show how to set up mirrors next to a path graph in order to make the connection between the ring and path graphs geometrically obvious. Solution: (a) The Laplacian for the path graph is L = 1 - 1 - 1 2 . . . - 1 . . . - 1 . . . 2 - 1 - 1 1

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For the ring graph, the only diﬀerence is that the ﬁrst and last vertices are neighbors, so L = 2 - 1 - 1 - 1 2 . . . - 1 . . . - 1 . . . 2 - 1 - 1 - 1 2 (b) If v R N is an eigenvector for the path graph, then L p v = λv If w = ( v 1 ,v 2 ,...,v n ,v n ,v n - 1 ,...,v 1 ) > , then ( L r v ) i = λv i for i = { 2 ,...,N - 1 ,N + 2 ,..., 2 N - 1 } since the ring graph Laplacian is the same on these entries as the path graph Laplacian, and the vectors are the same (up to a reversal of order on the last half of the vector, which doesn’t matter). On the ﬁrst coordinate, we know that since v is an eigenvector of L p , v 1 - v 2 = λv 1 , so ( L r w ) 1 = 2 w 1 - w 2 N - w 2 = 2 v 1 - v 2 - v 1 = v 1 - v 2 = λv 1 The argument is similar for the other endpoints, so w is an eigenvector of L r . Likewise, if
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hw4soln - CME 305: Discrete Mathematics and Algorithms...

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