Problem (2): Draw the complete graph on 5 vertices, Ks, and number each vertex 1 through 5. Weight each edge by the formula [], where v₁ and 2 are the numbers labelling the vertices with which the edge is incident. Use Kruskal's Algorithm to find a minimum spanning tree.

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8. Use Prim's algorithm and Kruskal's algorithm to find a minimum spanning tree for the weighted graph given below. 5 4 а 2 3 5 3 7 e d 61 3 4 4 6 2.

(1) T F Given a directed graph G and a vertex v in the graph, breath first search (BFS) can be used to detect if v is part of a cycle in the graph. (2) T F Let P be a shortest path from some vertex s to some other vertex t in a directed graph. If the weight of each edge in the graph is decreased by one, then P will still be a shortest path from s to t. (3) T F edge Kruskal's algorithm is always correct even in graphs with negative weights. (4) T F For any flow network, there is only one unique way to assign flow value to the edges so as to achieve the maximum flow for the network. NP problems are those problems that cannot be solved in polynomial (5) T F time.

Given the below graph, show the minimum spanning tree using an adjacency matrix. You may use scratch paper to draw the graph if you wish. 12 A В 2 3 E F 8 G H 10 A В D E F G A В C D E F G H |||

Problem 4: Consider the connected simple graph G given below. k Graph G 1. Use depth-first search to produce a spanning tree for the simple graph G. Choose vertex a as the root of this spanning tree. You must show how the spanning tree is constructed step by step as you add vertices and edges. Otherwise, your answer is wrong. Show your work step by step. 2. Use breadth-first search to produce a spanning tree for the simple graph G. Choose vertex a as the root of this spanning tree and assume that all vertices are ordered alphabetically. You must show how the spanning tree is constructed step by step as you add vertices and edges. Otherwise, your answer is wrong. Show your work step by step. 3. Is a spanning tree of a simple connected graph unique? Explain your answer clearly.

10 15 13 4 16 12 14 H Consider the graph given above. Use Kruskal's algorithm to find the minimum spanning tree. a. What is the total weight of the spanning tree? b. List the weights of the selected edges separated by commas in the order of selection.

(a) Let G be a simple undirected graph with 18 vertices and 53 edges such that the degreesof G are only 3 and 7. Suppose there are a vertices of degree 7 and b vertices ofdegree 3. Find a and b. To receive any credit for this problem you must write completesentences, explain all of your work, and not leave out any details. problem 1, continued(b) Recall that a graph G is said to be k-regular if and only if every vertex in G has degreek. Draw all 3-regular simple graphs with 12 edges (mutually non-isomorphic). Hint:there are six of them. To receive credit for this problem, you should explain, as well aspossible, why all of your graphs are mutually non-isomorphic.

4. Using Floyd's algorithm, compute the distance matrix for the weight directed graph defined by the following matrix: 0 4 -2] 6 2 3 -3 2 0 4 ∞o 5 0 Show the intermediate matrices after each iteration of the outermost loop.

Apply Prim's algorithm or Kruskal's algorithm or Borvuka's algorithm to find the minimum spanning tree for the following graph. Illustrate the algorithm stepwise while applying it on the graph. |1