2. Find a counterexample to the converse of Theorem 11.8. Theorem 11.8: Let G = (V, E) be a loop-free graph with |V | = n ≥ 2. If deg(x) +deg(y) ≥ n − 1 for all x, y ∈ V, x̸ = y, then G has a Hamilton path.
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2. Find a counterexample to the converse of Theorem 11.8.
Theorem 11.8: Let G = (V, E) be a loop-free graph with |V | = n ≥ 2. If deg(x) +
deg(y) ≥ n − 1 for all x, y ∈ V, x̸ = y, then G has a Hamilton path.
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- 2.4 (Ross, 2003, page 593) Show that the relationship $(x) = x;¢(1;, x) + (1 – x;)$(0;, x) is true for any coherent structure function o (x) and any i = 1, 2, ., n. ...95.- State A for the linear transformation T(ü) = Aŭ " (1) - |-| -c+d P 1 -1 a) A = 1 b) A = 1 -1 1 1 c) A = |-1 1 d) None of the above.8. Prove or disprove the following: (a) If X and Y are path-connected, then X x Y is path-connected. (b) If A C X is path-connected, then A is path-connected. (c) If X is locally path-connected, and AC X, then A is locally path-connected. (d) If X is path-connected, and f: X Y is continuous, then f(X) is path- connected. (e) If X is locally path-connected, and f: X→ Y is continuous, then f(X) is locally path-connected.
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- T. I need help with this discrete math problem! ONLY solve question 7! The other question serves as background information so you don't need to do it!3. Consider a formula Væ3y3zVuE(x, y) ^ E(x, z) ^ (u = x V ¬E(u, y) V ¬E(u, z)). Here, 2, y, z, u e V for some graph G = (V, E), and E() is the edge relation. Assume that Vu¬E(v, v), and that |V| = n. (a) What is the maximal possible number of edges in an undirected graph satisfying this formula? (b) What is the minimal possible number of edges in an directed graph satisfying this formula? Show both that there exists a graph with that many edges (you can just describe how to construct such a graph), and that this number is optimal.1. Let k E N. Let A be the set of all 2k-tuples (a1, ..., a2k) such that • ɑ; = 0 for k values of i, • ai = 1 for k values of i, and • for each i, Laj > j=1 (a) Find a bijection between A and some set B of shortest paths that you think you can count. Justify that your function is a bijection. (b) Find |B|.