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4. Graphs represented in Adjacency Matrices
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- In Computer Science a Graph is represented using an adjacency matrix. Ismatrix is a square matrix whose dimension is the total number of vertices.The following example shows the graphical representation of a graph with 5 vertices, its matrixof adjacency, degree of entry and exit of each vertex, that is, the total number ofarrows that enter or leave each vertex (verify in the image) and the loops of the graph, that issay the vertices that connect with themselvesTo program it, use Object Oriented Programming concepts (Classes, objects, attributes, methods), it can be in Java or in Python.-Declare a constant V with value 5-Declare a variable called Graph that is a VxV matrix of integers-Define a MENU procedure with the following textGRAPHS1. Create Graph2.Show Graph3. Adjacency between pairs4.Input degree5.Output degree6.Loops0.exit-Validate MENU so that it receives only valid options (from 0 to 6), otherwise send an error message and repeat the reading-Make the MENU call in the main…A directed graph G is defined using the following sets of vertices and edges. {n1, n2, n3, n4, N5, n6, n7, ng } E = {, , , , , , , , , } Q4) V = a) What is the adjacency matrix of G? b) List all semipaths of length 2 and 3. c) Give the condensation graph of G.2. Let G = (V, E) be a directed weighted graph with the vertices V = {A, B, C, D, E, F) and the edges E= {(A, B, 12), (A, D, 17), (B, C, 8), (B, D, 13), (B, E, 15), (B, F, 13), (C, E, 12), (C, F, 25)}, where the third components is the cost. (a) Write down the adjacency list representation the graph G = (V, E).
- 4. Let G=(V, E) be the following undirected graph: V = {1, 2, 3, 4, 5, 6, 7, 8} E = {(1, 7) (1, 4), (1, 6), (6, 4), (7, 5), (2, 5), (2, 4), (4, 3), (4, 8), (8, 3), (3, 5)}. a. Draw G. b. Is G connected? c. Give the adjacency matrix for the graph G given above. d. Determine whether the graph has a Hamiltonian cycle. If the graph does have a Hamiltonian cycle, specify the nodes of one. If it does not, prove that it does not have one. e. Determine whether the graph has a Euler cycle. If the graph does have a Euler cycle, specify the nodes of one. If it does not, prove that it does not have one.Write a Java program to find the Adjacency Matrix Representation using Directed Graph. а. Insert new nodes and directed edge between two nodes b. Display the representation1.6 If G is a simple graph of with number of vertices n > 0, show that deg(v) id(v) = od(v) = q. VEV VEV
- Let G = (V, E) be a directed graph, and let wv be the weight of vertex v for every v ∈ V . We say that a directed edgee = (u, v) is d-covered by a multi-set (a set that can contain elements more than one time) of vertices S if either u isin S at least once, or v is in S at least twice. The weight of a multi-set of vertices S is the sum of the weights of thevertices (where vertices that appear more than once, appear in the sum more than once).1. Write an IP that finds the multi-set S that d-cover all edges, and minimizes the weight.2. Write an LP that relaxes the IP.3. Describe a rounding scheme that guarantees a 2-approximation to the best multi-setConsider the graph in following. Suppose the nodes are stored in memory in a linear array DATA as follows: A, B, C, D, E, F, G, H, I, J, K, L, M Find the path matrix P of graph using powers of the adjacency matrix ALet G be a graph. We say that a set of vertices C form a vertex cover if every edge of G is incident to at least one vertex in C. We say that a set of vertices I form an independent set if no edge in G connects two vertices from I. For example, if G is the graph above, C = [b, d, e, f, g, h, j] is a vertex cover since each of the 20 edges in the graph has at least one endpoint in C, and I = = [a, c, i, k] is an independent set because none of these edges appear in the graph: ac, ai, ak, ci, ck, ik. 2a In the example above, notice that each vertex belongs to the vertex cover C or the independent set I. Do you think that this is a coincidence? 2b In the above graph, clearly explain why the maximum size of an independent set is 5. In other words, carefully explain why there does not exist an independent set with 6 or more vertices.
- (V, E) be an undirected graph. Assume that the vertices 2. Let G represent documents. The edges are weighted so that w(i, j) is the dissimilarity between documents i and j. It is desired to partition the vertices into k > 3 disjoint clusters such that k ΣΣ υ(u, υ). i=1 (u,v)EE u,vEC; is minimized. The set C; is the set of documents in cluster i. Show that the e-approximate version of this problem is P-hard for all e, e > 0. Note that k is a fixed integer provided with each problem instance and may be different for different instances.Discrete mathematics. Let G = (V, E) be a simple graph4 with n = |V| vertices, and let A be its adjacency matrix of dimension n × n. We want to count the L-cycles : such a cycle, denoted by C = u0u1 · · · uL with uL = u0 contains L distinct vertices u0, . . . , uL-1 et L edges E(C) = {uiui+1 | 0 ≤ i ≤ L − 1} ⊆ E. Two cycles are distinct if the edge sets are different : C = C' if and only if E(C) = E(C'). We define the matrices D, T, Q, the powers of A by matrix multiplication : D = A · A = A2, T = A · D = A3, Q = A · T = A4. Consider the values on the diagonals. Prove that for any vertex u ∈ V with degree d(u), d(u) = Du,u.Discrete mathematics. Let G = (V, E) be a simple graph4 with n = |V| vertices, and let A be its adjacency matrix of dimension n × n. We want to count the L-cycles : such a cycle, denoted by C = u0u1 · · · uL with uL = u0 contains L distinct vertices u0, . . . , uL-1 et L edges E(C) = {uiui+1 | 0 ≤ i ≤ L − 1} ⊆ E. Two cycles are distinct if the edge sets are different : C = C' if and only if E(C) = E(C'). We define the matrices D, T, Q, the powers of A by matrix multiplication : D = A · A = A2, T = A · D = A3, Q = A · T = A4. Consider the values on the diagonals. Prove that the number of 3-cycles N3 in the whole graph is N3 = 1/6 ∑ u∈V Tu,u