The representatives to the United Nations from different countries meet for a summit. After a few common sessions, the representatives have to participate in some one-on-one sessions with one another to discuss issues between the two countries. We have to schedule these one-on-one sessions into time slots and we would like as few time slots as possible for more manageable logistics. Model this problem as a graph coloring problem (State what are the vertices, what are the edges, what is being colored, what do the colors stand for).

Mark Zuckerberg, the CEO of Facebook, has hired you to lead the Facebook Algorithms Group. He has asked you to use various graph algorithms to analyze the world's largest social network. The Facebook Graph has 2.8 billion vertices, with each vertex being a Facebook user. Two vertices are connected provided those two users are "friends". The first decision you need to make is how you want to model the Facebook graph. Determine whether you should use an adjacency-list representation or an adjacency-matrix representation.

Submit solutions to the following problems in a single pdf document preferably completed using LaTeX. It is likely that you will need to use programming (Python) or a matrix calculator to solve most of the problems. You do not need to submit the code, but you should include explanations of all your conclusions. 3. Consider the cat polygon with vertices at (0, 0), (3, 3), (3, 6), (2, 7), (1, 6), (-1, 6), (-2, 7), (-3, 6), and (-3, 3). a. Write a single matrix that will make the cat's face twice as tall, half as wide, and rotate anti- clockwise 45 degrees. What is the determinant of the transformation matrix? b. Calculate the area of the cat's face prior to transformation and after the transformation. How does the change in area relate to the determinant of the transformation matrix?

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Problem 2. The two problems below can be solved using graph coloring. For each problem, represent the situation with a graph, say whether you should be coloring vertices or edges and why, and use the coloring to solve the problem. a. Your Quidditch league has 5 teams. You will play a tournament next week in which every team will play every other team once. Each team can play at most one match each day, but there is plenty of time in the day for multiple matches. What is the fewest number of days over which the tournament can take place? b. Ten members of Math Club are driving to a math conference in a neighboring state. However, some of these students have dated in the past, and things are still a little awkward. Each student lists which other students they refuse to share a car with; these conflicts are recorded in the table below. What is the fewest number of cars the club needs to make the trip? Do not worry about running out of seats, just avoid the conflicts. Student: A B C D E F…