Given the adjacency matrix of a graph: 0 20010 0 110 1 0 120 0 00 10 1 Answer the following questions. • What is the order of the graph? • How many parallel edges are there? • How many loops are there? • What is the sum of all degrees? For the following statement, type T if it is true or F if it is false. • The graph has an Eulerian path.
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- (it is algoritma and graph theory class.) If G is a graph without loops, what can you say about the sum of the entries in (i) any row or column of the adjacency matrix of G? (ii) any row of the incidence matrix of G? (iii) any column of the incidence matrix of G?Consider the adjacency matrix for a graph that is shown below. Answer the following questions by examining the matrix and its powers only, not by drawing the graph. Show your work in a way that makes your reasoning clear. (a) How many walks of length 2 are there from v1 to v3? (b) How many walks of length 3 are there from v1 to v2?Question 13given the adjacency matrices construct the corresponding graphs
- Draw a connected bipartite graph G = (V, E) such that its bipartition V1, V2 has |V1| = 5 and|V2| = 4 and deg(a) ≤ 3 for all a ∈ V .(a) Determine how many cut vertices and how many cut edges G has.(b) How many 1’s does the incidence matrix have in it?(c) Does your graph have an Euler path? Justify your answerIII. Consider the directed graph described by the following:(a) Draw the graph.(b) Find a directed path from vertex 3 to vertex 6.(c) Find a directed cycle starting from and ending at vertex 4.(d) Find the adjacency matrix of the graph.(e) Does there exist a directed path from vertex 2 to vertex 6?Which of the following is false? A.) Hamiltonian cycle can be converted to a Hamiltonian path by removing one of its edge. B.) Every graph that contains a Hamiltonian cycle also contains a Hamiltonian path and vice versa is true. C.) There may exist more than one Hamiltonian paths and Hamiltonian cycle in a graph. D.) A connected graph has as Euler trail if and only if it has at most two vertices of odd degree
- Which of the following are true for the adjacency matrix of the attached graph? 1. ||A||F = 4 2. A[1,.... ,1]T gives the vector of all out-degrees (number of edges out from a vertex) 3. ||A|| infiniti = 3 4. A is symmetricTheorem 3.5 states the following: Let G be a loopless graph with at least three vertices, and no isolated vertices. Then G is 2-connected if and only if, for every pair {e, f} of edges of G, there is a cycle of G that contains both e and f.Which of the following statements are true. Do not show your explanations. (d) Two graphs are isomorphic to each other if and only if they have the same adjacency matrix.(e) If T is a tree with e edges and n vertices, then e + 1 = n. (f) Petersen graph is not Hamiltonian graph.
- This question is related to Discrete Mathematics Consider the map of an Airport with wings: A-C and arms: a-o Construct a graph for it Build the Adjacency matrix for the graph Find Euler pathA rat is put into the following maze: The rat has a probability of 1/4 of starting in any compartment and suppose that the rat chooses a passageway at random when it makes a move from one compartment to another at each time. Let Xn be the compartment occupied by the rat after n moves. Explain why {Xn} is a MC and find the transition matrix P . Explain why the chain is irreducible, aperiodic and positive What is the limit of Pn? Find the probability that the rat is in compartment 3 after two In the long run, how many times that the rat enters in compartment 4 in 100 movements?Hey, The condensation of a graph G with k strong coherence components G1 =.(V1 , E1 ), . . . , Gk = (Vk , Ek )is the reduction of the original graphto its strong coherence components. In this case, the coherence components are combined into one node each in the condensation. The condensation to G is thus the graph G↓=({V1,...,Vk},E),where(Vi,Vj)∈E ⇔i̸=j∧∃u∈Vi,v∈Vj:(u,v)∈E holds. what is the Kondensation G↓ of the graph in the picture? Thank you in advance!