1. What is the order in which the vertices of the above ordered rooted tree are visited using preorder traversal? Show your work step by step. 2. What is the order in which the vertices of the above ordered rooted tree are visited using inorder traversal? Show your work step by step. 3. What is the order in which the vertices of the above ordered rooted tree are visited using postorder traversal? Show your work step by step. 1

6. Consider the following graph: 1 2 5 4 8 3 6 9 7 List the order in which the vertices are visited with a topological sort, as shown in class. Always process the vertex with the lowest number first if several vertices have indegree 0. 10

4. Consider the graph G, shown below, which is simply a copy of K5. 02 V3 5 V1 24 V5 How many distinct spanning trees does G have? (Hint: Break up your search by the isomorphism type of the tree, as discovered on the previous page. So for example, start by counting the paths of length 5 in G. Then proceed to the next type of tree with 5 vertices. The total number of trees is 125, but please use this answer only to check that your solution is complete!)

It was claimed that anyone involved in the Hollywood film industry can be linked through their film roles to Kevin Bacon within six steps. You are suspicious of such a claim and decide to verify it using the given dataset.1. Before you start to verify the claim, you first notice that it is not possible for this claim to be true if the actor graph is not even connected. If the given undirected graph is not connected, then some vertices must be in different connected components. Briefly explain how you can find the number of connected components in an undirected graph. Include the runtime of your algorithm as well.

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.

11. For the given graph below, use the depth-first search algorithm to visit the vertices and write the sequence of the visit. Start from the vertex "A". At any given vertex, you will visit first the unexplored vertex that has least alphabetical order. с G E F Answer the sequence of visiting the vertices: -_