Prove that for any two simple connected joint graphs G and H, L(G) UL(H) L(GUH).
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- al n+1 #2,2", Q4. (a) Prove that for any two simple connected joint graphs G and H, * L(G) UL(H) L(GUH). Aut (6) iff G contains atIn graphing, prove that at least one of G and it's complement, G', is connected.(b) Prove that Qn+1 Zy K₂" 2", for any integer n ≥ 2. Q4. (a) Prove that for any two simple connected joint graphs G and H, *
- Let G be a connected cubic graph of order n > 4 having clique number 3. Determine x(G).9. Which of the following statements are true? T(G) is connected for every connected graph G. If T (G) has more than one vertex then it has a cycle. If G and G'are isomorphic then so are T (G) and T(G'). □ IfT(G) and T (G') are isomorphic then G and G' are isomorphicLet H be any graph, and let H’ be the graph with V(H’) = V(H) U {v} for some vertex v not in V(H), and E(H’) = E(H). For each positive integer n and graph H, find ex(n, H’) in terms of ex(n, H).
- The parts (a) and (b) of this problem are independentof each other.G1 G24 51 236sx yt u v(a) Prove that the graphs G1 and G2 are isomorphic byexhibiting an isomorphism from one to the other byconcrete arguments and verify it by using adjacencymatrices. Please take the ordering of the vertices as1, 2, 3, 4, 5, 6 while forming AG1, adjacency matrix ofG1.Warning: One must stick to the labelings ofthe vertices as they are given, if one changesthe labelings/orderings etc., the solution willnot be taken into account.(b) Consider the complete graph K13 with vertex setV13 = {u1, u2, u3, · · · , u13}.Let H = (V, E) be the simple graph obtained fromK13 by adding a new vertex u, i.e. V = V13 ∪ {u}and deleting the edges {u1, u2} and {u2, u3} andadding the edges {u1, u} and {u, u2} and keepingthe remaining edges same.Determine whether H has an Euler circuit or not,an Euler path or not. One must validate any conclusion by clear arguments..-) oketen a graph of p(x) =-21x-11 + 4 2 12 6-5 -4 -3-21 1 2 3 456 2 4 0-3 5 6 7. Given f (x) x24, g(x) 3- x, Find the following: a) The domain of f(x); 2 C3) Cy) U Cty ) 3) The domain of g (x); b) 2 g) (x) The domain of d) Page 3 of 7Prove: Let F,F'be forests on the same set of vertices, with |E(F)|< |E(F')|. Show that F'has an edge e such that F + e is again a forest.
- Prove or disprove: There exist two connected graphs G and H both of order at least 3 and neither of which is Eulerian such that G + H is Eulerian.Let L(G) be the line digraph of a digraph G of order v≥ 2. Prove that (a) L(G) is strongly connected if and only if G is strongly connected; (b) L(G) ≈ G if and only if G is a directed cycle;Let G = (V, E) be a connected, undirected graph. Let A = V, B = {1,..., n − 1}, and ƒ(v) = deg(v). Select all that are true. a) f : A → B is not a function b) f: A → B is a function but we cannot always apply the Pigeonhole Principle with this A, B c) f: AB is a function but we cannot always apply the extended Pigeonhole Principle with this A, B d) none of the above
