For each definition of f below, determine if f is a function. If it is a function, state its domain and codomain in the form "f:X → Y" and give a table of values that lists each element x of the domain along with the corresponding element f (x) of the codomain f the given f is not a function, explain why not.
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PLEASE ANSWER ALL PARTS OF THIS QUESTION.
For the following directed graph, let ? = {?,?,?, ?, ?} and let ? = {?, ?, ?, ?}.
For the definition of ? below, determine if ? is a function. If it is a function, state its domain and codomain in the form “?:? → ?” and give a table of values that lists each element ? of the domain along with the corresponding element ��(?) of the codomain. If the given ? is not a function, explain why not.
If ? is a vertex, then ?(?) is the indegree of ?.
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- (V, E) be a connected, undirected graph. Let A = V, B = V, and f(u) = neighbours of u. Select all that are true. Let G = a) f: AB is not a function Ob) f: A B is a function but we cannot always apply the Pigeonhole Principle with this A, B Odf: A B is a function but we cannot always apply the extended Pigeonhole Principle with this A, B d) none of the aboveIn 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…5. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.) A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u. A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component. (Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.) Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…
- Consider the un-directed graph shown in the Fig. The values inside the node refer to the feature value of the node. The order of the nodes is A,B,C,D when following the matrix form in below questions.(You can use python or other languages to find the eigen values and other operationfor this question):1. For the given graph, write down the degree matrix, adjacency matrix, and Laplacian matrix representation.2.Find the eigenvalues and eigenvectors of the graph Laplacian matrix. Do you see any pattern pertaining to the signs of the eigenvalues (all positive or negative or no such pattern). Will it be same for all such graph Laplacianmatrices, and why?(a) Assume there is an adjacency matrix A of an undirected graph G. Further assume a 1 is indicated in the element, A(ij), if there is an edge between node i and node j, and a 0 is indicated in the element A(ij) if there is no edge between node i and node j. Explain the characteristics of the matrix A when (i) the graph is complete, (ii) the graph has a loop (edge connecting a vertex to itself), (iii) the graph has an isolated vertex, i.e., a vertex with no edges incident on it. (b) Repeat (a) for adjacency list representation, each row takes a linked list like structure.QUESTION 1: Given the following adjacency matrix representation of a graph to answer the followed questions. 0 11|0| 1 1001| 0 01 0 0 0100 | 1 1 1 | 0 1 1 D. reRepresent the graph using an adjacency list. E. Draw the original graph
- 1. Consider the directed acyclic graph (DAG) D shown below. (a) Write down the adjacency matrix A corresponding to the ordering of the vertices given by alphabetical order: a, b, c, d, e, f, g. (b) Find all permutations of the set of vertices {a, b, c, d, e, ƒ, g} such that the asso- ciated adjacency matrix is strictly lower triangular. (c) Find the number of spanning trees rooted at vertex g.c. Below an adjacency list representation of an undirected graph where there are no weights assigned to the edges. Draw 1. the graph and 2, the adjacency matrix with this adjacency list representation. 1 2 3 2 1 3 3 1 4 4 4 5 4 5 / 43) The graph k-coloring problem is stated as follows: Given an undirected graph G = (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in Va color c(v) such that 1< c(v)In order to plot a graph of f(x)=z and g(y)=z in the same graph, with t as a parameter. The function used is mesh(z) O subplot(x,y,z) O plot(x,y,z) O plot3(x,y,z) OThe Graph Data Structure is made up of nodes and edges. (A Tree Data Structure is a special kind of a Graph Data Structure). A Graph may be represented by an Adjacency Matrix or an Adjacency List. Through this exercise, you should be able to have a better grasp the Adjacency Matrix concept. You are expected to read about the Adjacency Matrix concept as well as the Adjacency List concept. Suppose the vertices A, B, C, D, E, F, G and H of a Graph are mapped to row and column indices(0,1,2,3,4,5,6,and 7) of a matrix (i.e. 2-dimensional array) as shown in the following table. Vertex of Graph Index in the 2-D Array Adjacency Matrix Representation of Graph A B 2 F 6. H 7 Suppose further, that the following is an Adjacency Matrix representing the Graph. 3 4 5. 6. 7 0. 1 1 1 1 01 1 01 1. 3 14 1 1 1 6. 1 Exercise: Show/Draw the Graph that is represented by the above Adjacency matrix. Upload the document that contains your result. (Filename: AdjacencyMatrixExercise.pdf) Notes: -The nodes of the…3) The graph k-coloring problem is stated as follows: Given an undirected graph G= (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in V a color c(v) such that 1SEE MORE QUESTIONSRecommended textbooks for youDatabase System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill EducationStarting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSONDigital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSONC How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSONDatabase Systems: Design, Implementation, & Manag…Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage LearningProgrammable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill EducationDatabase System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill EducationStarting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSONDigital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSONC How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSONDatabase Systems: Design, Implementation, & Manag…Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage LearningProgrammable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill Education