#20, 21, 23, 26
700 10/Graphs
.Construct the underlying undirected graph for the graph with directed edges in Figure 2. 12.What does the degree of a vertex represent in the acquain- tanceship graph. where vertices represent all the people in the worid? What does the neighborhood of a vertex in this graph represent? What do isolated and pendant ver- tices in this graph representIn one study it was estirsated that the average degree of a vertex in this graph is 1O00 What does this mean in terms of the model? 13. What does the degree of avertex represent in an academic collaboration graph? What does the neighborhood of a
represent?
14. What does the degree of a vertex in the Hollywood graph represent What does the neighborhood of a vertex rep- reseat? What do the isolated and pendanr vertices repre-
26.For which values of n are these graphs bipartite? a)K, c)W d 27.Suppose that there are four cmployees ia the computer support group of the School of Engincering of a large uni- versity.Each employec will be assigned to support one of four differeat areas: hardware,sofrware, nctworking. and wireless.Suppose that Ping is qualified to support hardware. networking,and wireless: Quiggley is quali- fed to support software and networkingRui is qualified to support networking and wireess nd Sitea is qualified to support hardware and software, a Use a bipartite graph to model the fouremployees and their qualifications b Use Hall's theorem to determine whether there is an assigrment of employees to support areas so that cach employee is assigned one area to support c If an assignment of cmployees to support areas so that each employee is assigned to one suppori area exists find one. 28.Suppose that a ncw compuny has five employces: Zamora. Agraharam Smith, Chou,and Macintyre. Each employec will assume one of six responsiblitieplan ning.publicity. sales, marketing. development and in- dustry relatiom.Each employec is capable of doing one or more of these jobs:Zamora could do planning.sales marketing. or industry relations Agraharam could do planning or development:Smith could do publicity.sales. or industry relatiomChou could do planning, sales or industry relations and Macintyre could do planning publicity,sales.or industry relations a Model the capabilities of these employees using a bi partite graph. b Find an assignment of responsibilites such that each employee is assigned one responsibility. c) Is the matching of responsibilities you found in part (b)a complete matching?Is it a maximum maiching?
IS. What do the in-degree and the out-degree of a vertex in a telephone call graph.as described in Example 4 of Sec tion 10.1.represent What does the degree of a vertex in the undirected version of this graph represent? 1. What do the in-degree and the out-degree of a vertex in the web graph.as described in Example 5 of Section 10.1 represeot? 17.What do the in-degree and the our-degree of a vertex in a directed graph modeling a round-robin tournament rep- resent? 18. Show that in a simple graph with at least two vertices there must be two vertices that have the same degree 19.Use Exercise 18 to show that ia a group of people. there must be two people who are friends with the same num- ber of other people in the group 20.Draw these graphs. K bKi K dC eW in Exercises 2125 deermine whether the graph is bipartite You may find it useful to apply Theorem 4 and answer the question by determining whether it is possible to assign ei- ther red or blue to each vertex so that no two adjacent vertices are assigned the same color.
21.2
29. Suppose tha there are five young women and five young men on an island. Each man is willing to marry some of the women on the island and cach woman is willing to