Use Dirac's Theorem to verify that the graph is hamiltonian and then find a hamiltonian circuit.
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Use Dirac's Theorem to verify that the graph is hamiltonian and then find a hamiltonian circuit.
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- What does Dirac’s Theorem state? Explain how it guarantees that the graph is HamiltonianWhat is the number of Hamiltonian paths in Kn+1, n? Justify your answer.Use trial and error to find two Hamiltonian circuits of different total weights, starting at vertex A in the weighted graph. Compute the total weight of each circuit. (Select all that apply.) E 'D O A-B-E-D-C-A, 32 O A-B-E-D-C-A, 31 O A-D-E-B-C-A, 31 О -С-D-E-B-А, 32 O A-D-E-B-C-A, 32
- The picture on the left is that of an icosahedron, a solid object whose faces consist of 20 congruent equilateral triangles. By stretching the base triangle and flattening, the icosahedron determines a graph in the plane (as shown on the right side of the figure). Find a Hamiltonian cycle in this graph.Transcribe the hamiltonian onto the Heisenberg square H = 2a2. Answer the following questions. a. If a graph is Hamiltonian, is it necessarily Eulerian as well? If yes, explain why. If no, provide a counterexample. b. If a graph is Eulerian, is it necessarily Hamiltonian as well? If yes, explain why. If no, provide a counterexample.
- Prove that every hamiltonian digraph is strongly connected.2. Consider a connected network of two individuals. At each period t, each individual forms beliefs as a weighted average of his own beliefs with weight p = [0, 1] and the other agent's beliefs with weight 1 - p. (d) If p = 0, what happens to beliefs of each player as t→∞? (e) Express b2, beliefs in period 2, as a function of initial beliefs bo and p. Comment on the relationship between p and the difference in beliefs. (f) (Harder) What value of p will cause beliefs to converge fastest? Relate this to the second eigenvalue of the updating matrix A (Hint: your answer to (e) is helpful here.)" determine conditions on the bi's, if any, in order to guarantee that the linear system is consistent."
- Create a hermitian martix and show that it is a linear transformation.(b) Is it necessarily the case that for a graph with n vertices and exactly (₂¹)+1 edges be Hamiltonian? Either show this must be true of provide a counterex- ample.The canonical transformation b= ua + va" casts the Hamiltonian H[a, a'] wa'a+rla² + a°*] in the diagonal form H= Eb'b. Which system of linear equations relates u, v and E ? Select one: 1. Eu = wu - yo, Ev =-wv + yu. Eu =wu - 2yv, O 2. Ev = -wv +2yu. Eu =wv - yu, O3. Ev = -wu + yv. Eu =wv- 2-yu, O4. Ev =-wu + 2yv. O5. Eu = -wu + 2yv, Ev = wu - 2yu.