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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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 Spring '09
 M.PEI
 Math, Combinatorics, Natural number, Prime number, nonhomogeneous recurrence cn

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