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

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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 1s, where two vertices are adjacent if and only if the corresponding strings dier 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, dene 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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This note was uploaded on 10/23/2009 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.

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

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