Chapter 5 Review. 1. Write a summary of what you think are the important points of this chapter.

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1 DM_Final.qxp:DM.qxp // 8: AM Page 89 Chapter Review. Write a summary of what you think are the important points of this chapter.. Is the following graph planar or nonplanar? If it is planar, redraw it without edge crossings.. Show that the graph described by the following adjacency matrix is planar. A B C D E A B C D E. What is the chromatic number for each of the following? a. any tree with five vertices b. any tree with an odd number of vertices c. any tree with an even number of vertices d. any tree with two or more vertices

2 9 Chapter More Graphs, Subgraphs, and Trees. a. Explain why this graph is a bipartite graph. b. Is this graph a complete bipartite graph? Explain why or why not. c. Is this graph planar? If so, find a planar drawing for the graph. d. What is the chromatic number for this graph? 6. For the following graph, draw a subgraph that has vertices and edges. R S T U W V 7. Mr. Gonzalez, the principal at Central High School, leaves his office once an hour to visit the math, science, and civics classrooms, and then returns to his office. The distances between rooms are shown on the following graph. Science Office 6 ft 96 ft ft Math ft ft ft Civics

3 Chapter Review 9 a. Find the shortest route possible for Mr. Gonzalez. b. What is the total distance of the shortest route in part a? c. What kind of circuit does Mr. Gonzalez make? 8. For the following complete graph, find the total weight of the nearest-neighbor route starting at A. B A C 9 8 E D 9. Find a spanning tree for the following graph if one exists.. How many different spanning trees are there for a cycle with vertices? With vertices? With vertices? With n vertices?. For the following graph, explain why the darkened edges are not a spanning tree.

4 9 Chapter More Graphs, Subgraphs, and Trees. When given the position where it is currently located and its destination, a certain robot car is programmed to find the shortest path for the trip. The routes that the car can travel are shown on the following graph. Big Burger Pool Grocery Theater miles 7 Library Hospital 6 Home School Cleaners 6 a. Use inspection to find the shortest path from Home to Big Burger. b. What is the minimum distance from Home to Big Burger? c. Use the shortest path algorithm (page 8) to find the shortest paths from Home to each of the other locations on the graph.. Assume that all locations represented by the graph in Exercise need to be connected by cable. Find the minimum amount of cable needed to link the nine locations.. Are the following graphs trees? Explain why or why not. a. b.

5 Chapter Review 9 c. d.. What happens to a tree if the number of edges is increased by one? 6. Use the breadth-first search algorithm from page 6 to find a spanning tree for the following graph. Begin the algorithm at the vertex labeled S. S 7. The vertices of the following graph represent buildings on a small college campus. Administrators at the campus want to connect the buildings with fiber-optic cable and are interested in finding the least expensive way of doing so. The costs of connecting buildings (in thousands of dollars) are shown as weighted edges of the graph. 9 6 a. Use one of the spanning tree algorithms to find a minimum spanning tree for the graph. b. What is the total cost of connecting the buildings? 8

6 9 Chapter More Graphs, Subgraphs, and Trees 8. Draw a tree with eight vertices that has exactly four vertices of degree. 9. Create a problem that can be solved by using one of the minimum spanning tree algorithms and find the solution to your problem. Then give your problem to a classmate and ask him or her to solve it. If the answer differs from yours, determine the correct solution.. You roll a die with faces numbered 6. Then you flip a coin. Draw a tree diagram to show the possible outcomes.. Represent ( ) * 8 + as a binary expression tree.. Find a postorder listing for the following binary tree. + + * 6. Evaluate the following reverse Polish notation. 7 + * +. Create a mathematical expression of your own, represent it with an expression tree, and find the postorder listing for the tree. Give your listing to another student in your class and have him or her evaluate the reverse Polish notation. Check the value of the notation with the value of your original mathematical expression.

7 Chapter Review 9 Bibliography Chartrand, Gary. 98. Introductory Graph Theory. Mineola, NY: Dover Publications, Inc. Chavey, Darrah. 99. Drawing Pictures with One Line. (HistoMAP Module ). Lexington, MA: COMAP, Inc. COMAP.. For All Practical Purposes: Mathematical Literacy in Today s World. 9th ed. New York: W. H. Freeman. Cook, William J.. In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation. Princeton, NJ: Princeton University Press. Copes, W., C. Sloyer, R. Stark, and W. Sacco Graph Theory: Euler s Rich Legacy. Providence, RI: Janson. Cozzens, Margaret B., and R. Porter Mathematics and Its Applications. Lexington, MA: D. C. Heath and Company. Cozzens, Margaret B., and R. Porter Problem Solving Using Graphs. Lexington, MA: COMAP, Inc. Crisler, Nancy, and Walter Meyer. 99. Shortest Paths. Lexington, MA: COMAP, Inc. Dossey, John, A. Otto, L. Spence, and C. Vanden Eynden. 6. Discrete Mathematics. th ed. Upper Saddle River, NJ: Pearson. Francis, Richard L The Mathematician s Coloring Book. (HiMAP Module ). Lexington, MA: COMAP, Inc. Ore, O. 99. Graphs and Their Uses. Washington, DC: Mathematical Association of America. Queen Mary, University of London. "How bumblebees tackle the traveling salesman problem." ScienceDaily, 9 Jun.. Web. 8 Feb..