Given the following relations S and U, use ordered pair notation to express the relation U o S. b a b d d U US={Ex: (a, b), (b, c) S Hint: Start with a pair of vertices such as (a, b). If there exists some intermediate vertex that a can reach through one arrow in S and that intermediate vertex can reach b through one arrow in U, then (a, b) is in U o S. Otherwise, (a, b) is not in U o S.
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- Additional Topics: Topological Sort A strict order relation can be represented by a directed acyclic graph (or DAG), which is a directed graph with no cycles. For example, the following directed graph is a DAG: A topological sort of a DAG is an ordering of the vertices that is consistent with the edges of the graph. That is, if there is an edge (u, v) in the graph, then u should appear before v in the topological sort. For example, 1,2,3 and 1,3, 2 are topological sorts of the DAG shown above, but 2,1,3 is not a topological sort because 2 cannot be listed before 1. 7. Given the following DAG G representing UIC's Computer Science courses and prerequisites (note that edges implied by the transitive property are omitted): CS 261 CS 362 CS 141 CS 11 CS 211 Cs 361 CS 401 CS 151 CS 251 CS 342 Cs 301 Answer the following questions: a. Give two different topological sorts of G. b. Is each of the following orderings a topological sort of G? Justify your answer. CS111, CS151, CS301, CS251,…Assignment on Graph A social graph contains all the friendship relations (edges) among a group of n people (vertices). The friendship relationship is symmetric. Two vertices with no edge between them are enemies. Design and implement a class called Graph for a social graph. It should have an adjacency matrix as a data member to represent the graph. The nodes in the graph are numbered from 0..n-1 graph[i][i] - is always true (1) graph[i][j] is true if i and j are friends and false (0) if they are enemies. • In the constructor, initialize all entries to false (0) and all graph[i][i] to true (1) Include the following member functions. o void makeFriends(int i,intj)- make i and j friends int countFriends(int i) - will return the number of friends of i, excluding itself int countEnemies(int i)- will return the number of enemies of i int countCommonFriends(int i,intj) - the number of common friends of i and j int countCommonEnemies(int i,intj) - the number of common enemies of i and j int…A quadratic function f(x) is hidden from view. You must find the minimum value of the function f(x). Choose the form of the quadratic function f(x) that you would like to see in order to answer the question most efficiently. Form: Vertex Form ,f(x)=(x-3)^(2)-25
- Markup the BFS code so that it can determine if the graph g is connected or not. Further modify the BFS code so that that it is possible to determine the number of connected components and their membership (i.e., which vertices belong to which component). // Connectivity Checker and Identifier public static Integer[] ProcessOrderInBFS(Graph g) { int foundCount = 1; Integer[] processOrder = new Integer[g.size(O]; for (int i = 0; i q = new LinkedList(); int currNode = 0; Integer[] adjVector = null; q. add(startVertex); processorder[startVertex] = foundCount; foundCount++; while(!q.isEmpty()) { currNode = q.remove(); adjVector = g.getAdjVector(currNode); for (int i = 0; i < g.size(); i++) { if(adjVector[i] == 1 && processOrder[i] == 0) { q. add(i); processorder[i] = foundCount; foundCount++; } } } return foundCount;Specifications: You will create an implementation of this algorithm. Your driver program should provide a graph and a source vertex in the graph. Your implementation should use Dijkstra's Algorithm to determine the shortest path using adjacency matrix representation. Specifically, given a graph and a source vertex in the graph, find the shortest paths from source to all vertices in the given graph, using Dijkstra's Algorithm.The 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…
- 0 Give me examples of Suns graph Suns : Recall that a graph is chordal if it contains no induced cycles of length at least 4. A sun Sn is a chordal graph on 2n vertices (n3) whose vertex set can be partitioned into sets W = {w1, w2,...,wn} and U = {u1, u2,...,un} such that W is independent and ui is adjacent to wj if and only if i=jor i =j+1 (mod n) If U induces a complete graph then the graph is called the complete sun. 712 (Sn) = n. Proposition: For n>3,True or false helpConsider the following graph. 2 1) 4 Visually show how this graph will be saved in an adjacency matrix. Visually show how this graph will be saved in an adjacency list. Note: You only have to show the final adjacency matrix and adjacency list. No code required. 3.
- 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. The chromatic number of a graph is the least mumber of colors required to do a coloring of a graph. Example Here in this graph the chromatic number is 3 since we used 3 colors The degree of a vertex v in a graph (without loops) is the number of edges at v. If there are loops at v each loop contributes 2 to the valence of v. A graph is connected if for any pair of vertices u and v one can get from u to v by moving along the edges of the graph. Such routes that move along edges are known by different names: edge progressions, paths, simple paths, walks, trails, circuits, cycles, etc. a. Write down the degree of the 16 vertices in the graph below: 14…Assume you are to write a program to analyze the social connection between students in MTSU. Each student is a node in a undirected graph. An edge is added between two nodes if these two students have close social connections, i.e., in the same club, or in the same department. Which would be a better representation for this graph? Question 45 options: adjacency matrix representation adjacency list representationalgorithm GenericSearch (G, s) pre-cond: G is a (directed or undirected) graph, and s is one of its nodes. post-cond: The output consists of all the nodes u that are reachable by a path in G from s.