1. The graph G is given by the picture12456.89.1011(a) Explain why the transposition f = (5, 7) (swapping the vertices 5 and 7 andleaving all the other vertices fixed) and the product of transpositionsg = (1, 3)(4,8)(9, 11) define isomorphisms of G with itself.(b) Explain why the graph G is not Hamiltonian.(c) Find all pairs {v, w} of non-adjacent vertices v and w such that the graphG+ {v, w} is Hamiltonian. Justify your answer. [Hint: use the symmetriesof G given in part (a) to minimise the number of cases to consider.]

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Answered: 1. The graph G is given by the picture…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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4. Answer the following questions for the two graphs below. 15 14 13 Figur 1: G1 Figur 2: G2 (a) State the order and size of each graph. (b) Are the two graphs isomorphic? If yes, define the isomorphism and verify that each edge is preserved. If no, explain why. (c) What is the deg(r1) and deg(t)?
5.1.7. (-) Construct a graph G that is neither a clique nor an odd cycle but has a vertex ordering relative to which greedy coloring uses A(G) + 1 colors.
10.2.11. Let G = D34 = (a,b | a¹7 = b² = e,ba = a¹6b). Let H = (a) and K = = (b). (a) Draw a partial lattice diagram of subgroups of G that includes the subgroups H and K. Include also H¦ K and (H,K). Label the edges with appropriate numbers, and give reasons for what you have done. In what follows, you may refer back to your diagram. (b) Is H
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Consider the given graphs: U18 u₂ 16 15 ●113 14 Vg V7 V₁ V6 V/₂ V5 V3 V4 G H Click and drag the statements in correct order either to show that these graphs are not isomorphic or to find an isomorphism between them using paths. Proceeding along similar lines, complete the bijection with us → V8, U6 → V7, and U7 → VS. Having thus been led to the only possible isomorphism, check that the 12 edges of G exactly correspond to the 12 edges of H, and the two graphs are isomorphic. Now u, is adjacent to uz and us, and v₂ is adjacent to v₁ and vi, so u₂ v. and us > V Having thus been led to the only possible isomorphism, check that the 12 edges of G do not correspond to the 12 edges of H; hence the two graphs are not isomorphic. Since u4 is the other vertex adjacent to us and v4 is the other vertex adjacent to V3, U4 → V4. So try to construct an isomorphism that matches them, say u₁ → V₂ and Us → V6.
5.1.21. Suppose that every edge of a graph G appears in at most one cycle. Prove that every block of G is an edge, a cycle, or an isolated vertex. Use this to prove that x (G) < 3.
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Abstract Algebra
Question 1: ( Are the following directed graphs strongly connected or not? Explain why in detail? 3 2 A В
4. Let G = (V, E) be a directed graph and let S1,. , Sm be its strongly connected components. Define a new graph G' = (V', E') as follows. V'= {1,.….. , m}. Place an edge from i to j in G' if there is a vertex u E S; and a vertex v E S; such that (u, v) E E. • Show that G' is a DAG. • Let n1, n2, ...Nm be a topological ordering of vertices in G'. Suppose that there is no edge between n; and ni+1 in G', then show that for every pair of vertices u, v, with u E S; and v E Si+1, there is no path from u to v in G.

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1. The graph G is given by the picture 1 4 5 6. 7 8 9. 11 10 (a) Explain why the transposition f = (5,7) (swapping the vertices 5 and 7 and leaving all the other vertices fixed) and the product of transpositions g = (1, 3)(4, 8)(9, 11) define isomorphisms of G with itself. 3.