B) Previously, we have learned how to represent a graph using the Adjacency matrix. Now in this problem, using the information in Task 01 (a), you have to build the graph using Adjacency List.
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- One can manually count path lengths in a graph using adjacency matrices. Using the simple example below, produces the following adjacency matrix: A B A 1 1 B 1 0 This matrix means that given two vertices A and B in the graph above, there is a connection from A back to itself, and a two-way connection from A to B. To count the number of paths of length one, or direct connections in the graph, all one must do is count the number of 1s in the graph, three in this case, represented in letter notation as AA, AB, and BA. AA means that the connection starts and ends at A, AB means it starts at A and ends at B, and so on. However, counting the number of two-hop paths is a little more involved. The possibilities are AAA, ABA, and BAB, AAB, and BAA, making a total of five 2-hop paths. The 3-hop paths starting from A would be AAAA, AAAB, AABA, ABAA, and ABAB. Starting from B, the 3-hop paths are BAAA, BAAB, and BABA. Altogether, that would be eight 3-hop paths within this graph. Write a program…In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. in its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices share the same color; this is called a vertex coloring.", Wikipedia a) Apply your algorithm in the graph below. What is the accuracy ratio? b) Show that the performance ratio of your algorithm can go unbounded above or never at all."For the undirected graph shown below, give the number of vertices, the number of edges, and the degree of each vertex, and represent the graph with an adjacency matrix." This task is solved here, but it is only solved for task a, not b. could you help me with task b?
- Suppose you have a graph G with 6 vertices and 7 edges, and you are given the following information: The degree of vertex 1 is 3. The degree of vertex 2 is 4. The degree of vertex 3 is 2. The degree of vertex 4 is 3. The degree of vertex 5 is 2. The degree of vertex 6 is 2. What is the minimum possible number of cycles in the graph G?In hill-climbing algorithms there are steps that make lots of progress and steps that make very little progress. For example, the first iteration on the input given might find a path through the augmentation graph through which a flow of 30 can be added. It might, however, find the path through which only a flow of 2 can be added. How bad might the running time be when the computation is unlucky enough to always take the worst legal step allowed by the algorithm? Start by taking the step that increases the flow by 2 for the input. Then continue to take the worst possible step. You could draw out each and every step, but it is better to use this opportunity to use loop invariants. What does the flow look like after i iterations? Repeat this process on the same graph except that the four edges forming the square now have capacities 1,000,000,000,000,000 and the crossover edge has capacity 1. (Also move t to c or give that last edge a large capacity.)1. What is the worst case number of…*Discrete Math In the graph above, let ε = {2, 3}, Let G−ε be the graph that is obtained from G by deleting the edge {2,3}. Let G∗ be the graph that is obtain from G − ε by merging 2 and 3 into a single vertex w. (As in the notes, v is adjacent to w in the new if and only if either {2,v} or {3,v is an edge of G.) (a) Draw G − ε and calculate its chromatic polynomial. (b) Give an example of a vertex coloring that is proper for G − ε, but not for G. (c) Explain, in own words, why no coloring can be proper for G but not proper for G − ε. (d) Draw G∗ and calculate its chromatic polynomial. (e) Verify that, for this example,PG(k) = PG−ε(k) − PG∗ (k).
- c++ Given a collection of n edges, each represented by three numbers a, b, and c, you need to: • Output whether we can even make a MST. This requires that our graph can have all nodes connected to one another without cycling and without creating disjoint graphs. • Output the magnitude of the MST, or in other words the value of all edges of the MST added together. If there is no MST possible, output "No". If there is, output "Yes" and continue to the second task. On a new line, output "MST = " + the magnitude of the MST. a is the label of one vertex, b is the label of the other vertex, and c is the size of the edge that connects them. Each value is separated by a space, and each line ends with a endl character. The input will begin with the value of n, followed by a number of combinations of a, b, and c that could be constructed (note: the number of lines are not necessarily equal to or less than n). Make sure your output follows the structure given in the test cases. You MAY use STL…Create the ErdosRenyiGraph programme, which creates a graph by generating E random pairs of integers between 0 and V1 and using the command line inputs V and E to construct the graph. Self-loops and parallel lines are produced by this generator.Use of Matching for Edge ColouringEdges in a matching can share the same colour since matching in a graph is defined as the collection of non-adjacent edges. As a result, we can create an algorithm based on this idea using the below stages. 1. Input: G = (V, E)2. Output: Minimal edge coloring of G3. color ← 04. while G = ∅$5. find maximal matching M of G6. color all M vertices with color7. G ← G − M8. color ← color + 1implementation of Python
- An adjacency matrix might be a better choice for speeding up a program, however, it consumes huge memory for large graphs. How this situation can be improved? What programming constructs better suit graph representation? Explain with exampleI have to construct a connected graph ( some of the vertices have to connect two or more edges) from a list of pseudorandom numbers. I have tried this but got errors. Any help would be highly appreciated. import java.util.ArrayList; import java.util.List; import java.util.Random; public class MuddyTown { int weight; int m = 5; int d = 1; int k = 32; int s = 1; ArrayList<Integer> randomList = new ArrayList<>(); //ArrayList<Integer> randomList = new ArrayList<>(); //static ArrayList<Integer> randomList; ArrayList<String> connectedGraph = new ArrayList<>(); // Maximum number of vertices int V; //ArrayList<Integer>[] adj; List<List<Integer>> adjList; public MuddyTown() { V = randomList.size(); adjList = new ArrayList<>(V); for (int i = 0; i< V; i++) { adjList.add(new ArrayList<>()); } // for (int i = 0; i<V; i++) { // adjList.add(new ArrayList<>(V)); // } } ArrayList<Integer> pseudoRandom() { for…Write a program (WAP) to create an undirected graph using adjacency matrix representation.Number of nodes and edges should be taken from the user. After creating the graph, performfollowing operations: (i) Search a node. Take the node number from the user. If the node is found then print its associatededges.(ii) Insert a node in the graph.(iii) Insert an edge in the graph. Take the node numbers from the user between which the edge is tobe inserted.(iv) Delete a node from the graph. Take the node number to be deleted from the user.(v) Apply DFS on the graph and print the graph traversal.(vi) Apply BFS on the graph and print the graph traversal.Solve the above problem using adjacency list representation.