Hw7solns - Discrete Mathematics Spring 2004 Homework 7 Sample Solutions 6.1#41 How many edges are in an n-cube Solution For concreteness lets use

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Discrete Mathematics, Spring 2004 Homework 7 Sample Solutions 6.1 #41. How many edges are in an n -cube? Solution . For concreteness, let’s use the bit string representation of the n -cube, in which vertices are connected by an edge if their corresponding strings di±er in exactly one position. Let b n denote the number of edges in the n -cube. Then the ²rst few cases can be done manually: b 1 = 1, b 2 = 4, b 3 = 12. In general, one should notice that the n -cube is obtained from the ( n - 1)-cube in the following manner: take two copies of the ( n - 1)-cube (one copy consisting of strings which start with a 0, the other consisting of strings which start with a 1), and connect with an edge those vertices whose strings match in bits 2 through n but di±er in the ²rst bit. Thus we are adding 2 n 1 new edges, one for each vertex of the ( n - 1)-cube. From this it follows that the recurrence relation for the sequence b n follows the pattern b n = 2 b n 1 + 2 n 1 , and we can solve by iteration: b n = 2 b n 1 + 2 n 1 = 2(2 b n 2 + 2 n 2 ) + 2 n 1 = 2 2 b n 2 + 2 · 2 n 1 = 2 2 (2 b n 3 + 2 n 3 ) + 2 · 2 n 1 = 2 3 b n 3 + 3 · 2 n 1 = ··· = 2 n 1 b 1 + ( n - 1) · 2 n 1 = n · 2 n 1 . Note: one can use an alternate approach based on the methods of section 2. Notice that each vertex has degree n , and there are 2 n vertices in all. From Theorem 6.2.21 we know that the sum of the degrees of the vertices is 2 b n , we conclude that b n = 1 2 (number of vertices)(degree of each vertex) = 1 2 · 2 n · n = 2 n 1 · n. 1
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6.2 #15. Draw a graph with four vertices having degree 1, 2, 3, and 4, or explain why no such graph exists. Solution . There are a number of possible graphs satisfying these criteria. The following is one possibility: 6.2 #18. Draw a simple graph with Fve vertices having degrees 2, 2, 4, 4, and 4, or explain why no such graph exists. Solution
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This note was uploaded on 06/12/2011 for the course MATH 103 taught by Professor Wouters during the Spring '08 term at Wisc Oshkosh.

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Hw7solns - Discrete Mathematics Spring 2004 Homework 7 Sample Solutions 6.1#41 How many edges are in an n-cube Solution For concreteness lets use

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