sol12 - CS 330 Discrete Computational Structures Fall...

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CS 330 : Discrete Computational Structures Fall Semester, 2014 Assignment #12 [Extra Credit] Due Date: Friday, Dec 12 Suggested Reading: Chapter 11.1 - 11.3 and 11.9 - 11.11 of Lehman et al. These are the problems that you need to turn in. Always explain your answers and show your reasoning. Spend time giving a complete solution. You will be graded based on how well you explain your answers. Just correct answers will not be enough! 1. [10 Pts] Zhenbi Hu Give an example of a graph that has 6 vertices with degrees of 5, 3, 3, 3, 2, 2. How many edges does your graph have? Can you argue that any graph with these properties would have the same number of edges? Figure 1: Node a has degree 5; b, c, d has degree 3; e, f has degree 2. Solution: See Figure 1 for a specific example. The sum of the degrees is 5+3+3+3+2+2 = 18 . An edge must run between two vertices, of course, say v and w . That single edge accounts for a degree of 1 in v and a degree of 1 in w , so the number of edges will be 1 / 2 of the sum of degrees of the vertices. Therefore, any other graph with this degree sequence will also have 9 edges. We can also prove this immediately by invoking the Handshaking Theorem, which says that the number of edges is half the total vertex degree. 2. [10 Pts] Peter If G is a simple graph with n vertices and n - 1 edges, is G connected? If yes , give a short justification. If no , give a counterexample. Solution: No. The graph with 4 vertices { a, b, c, d } and 3 edges { ( a, b ) , ( b, c ) , ( a, c ) } is not connected.
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  • Fall '15
  • Soma Chaudhuri
  • Graph Theory, Vertex, vertices, G0

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