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# ass4 - n let G n be the graph whose vertices are all binary...

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MATH 239 Assignment 4 This assignment is due at noon on Friday, June 26, 2009, in the drop boxes opposite the Tutorial Centre, MC 4067. 1. You are given that x 3 - 3 x + 2 = ( x - 1) 2 ( x + 2) . (a) Find c n explicitly where c n satisfes the nonhomogeneous recurrence c n - 3 c n - 2 + 2 c n - 3 = - 2 n ±or n 3 , with initial conditions c 0 = - 1 ,c 1 = 7 ,c 2 = - 5. (b) Find c n explicitly where c n satisfes the nonhomogeneous recurrence c n - 3 c n - 2 + 2 c n - 3 = 6 ±or n 3 , with initial conditions c 0 = 2 ,c 1 = 5 ,c 2 = 1. 2. (a) Show that the ±ollowing two graphs are isomorphic. A B C D E F G H G 1 1 2 3 4 5 6 7 8 G 2 Figure 1: Two 3-regular graphs on 8 vertices (b) Show that the ±ollowing two graphs are isomorphic. A B C D E F G H G 1 1 2 3 4 5 6 7 8 G 2 Figure 2: Two 3-regular graphs on 8 vertices 1

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(c) Make a list of all 3-regular graphs with 8 vertices, up to isomorphism. In other words, each 3-regular graph with 8 vertices should be isomorphic to exactly one of the graphs on your list. BrieFy explain why no two are isomorphic. (Hint: there are exactly 6 graphs in the list.) 3. ±or any positive integer
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Unformatted text preview: n , let G n be the graph whose vertices are all binary strings of length n that have at most one block of 1’s, where two vertices are adjacent if and only if the corresponding strings di²er in exactly one position. (a) Determine the number of vertices p in G n . (b) Draw graphs G 3 and G 4 . (c) Determine the number of edges q in G n . 4. ±or n a positive integer, de³ne the prime graph P n as follows: V ( P n ) = { 1 , 2 ,... ,n } and E ( P n ) = {{ u,v } ⊆ { 1 , 2 ,... ,n } | u + v is a prime } . (a) Prove that P n is bipartite for all n . (b) Prove that P n is connected for all n . (You may assume, without proof, that for every integer k > 1, there is a prime number r such that k < r < 2 k .) 5. Prove that every graph with at least 2 vertices has two vertices of the same degree. 2...
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ass4 - n let G n be the graph whose vertices are all binary...

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