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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- 10. Let G = (V, E) be a loop-free connected undirected graph where V = {v1, v2, v3, . . . , vn},n ≥ 2, deg(v1) = 1, and deg(vi) ≥ 2 for 2 ≤ i ≤ n. Prove that G must have a cycle.2. Find a counterexample to the converse of Theorem 11.8 and show the full answer. 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.8. Let G be the graph below. (а) Does G has the Hamiltonian path? (b) Does G has the Hamiltonian cycle?
- 12. (1, 2). Find a formula for T(a1x + ao), where a1x + ao E P1(R). A linear map T: P(R)· - R? satisfies T(x) = (1, –1) and T(1)4) Use the superposition principal to draw a graph of each. a) y = f(x)+ g(x) b) y = g(x)-f(x) f(x) -4 g(x) f(x) YA 4 2 0 2 4 y 4 2 -4 0 A 2 g(x) -4 2 2 4x 4 x1. Show that y₁= ez and y2 = ze form a linearly independent set on the interval (-00,00).
- 2. 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.5.- 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.Let g(x) = e · (x² – 9). g' (x) = e* - (x² + 2x – 9), and g" (x) = e* · (x² + 4x - 7), Choose one for each part. a) When e · (x² + 2x – 9) 0, the graph of g(x) is decreasing concave up concave down c) e · (x2 + 4x – 7) < 0 on approximately [Select ] and therefore the graph of g'(x) is [ Select ] on this interval.
- 6. The covariance between X and Y is calculated and recorded as SXY (and sxy is a negative number). A linear transformation is performed on X. The linearly transformed variable is called W, where W; = 5 + Xi. The covariance between W and Y is calculated and recorded as sWY. Which statement is correct? A) В) In this example, swy would be less than sxY. In this example, swY Would be equal to sxY. In this example, swy would be greater than SXXY. 7. The covariance between X and Y is calculated and recorded as sxy (and sxy is a positive number). A linear transformation is performed on X. The linearly transformed variable is called W, where W; = (4)X;. The covariance between W and Y is calculated and recorded as sWY. Which statement is correct? In this example, sxY would be less than swy. In this example, sxY Would be equal to swy. In this example, sxY would be greater than sWY. A) B) 8. The correlation between X and Y is calculated and recorded as rxY (and rxy is positive). A linear…1.3.2 Determine which of the following functions are and are not linear forms. *(a) The function f : R" → R defined by f(v) = ||v||. (b) The function f: F"F defined by f(v) =v1. %3D *(c) The function f: M2 M2 defined by chip f b. a] %3D d [d b. C (d) The determinant of a matrix (i.e., the function det: Mn →F). *(e) The function g : P → R defined by g(f)= f'(3), where f' is the derivative of f. (f) The functiong:C → R defined by g(f) = cos(f(0)).14. Suppose we have a diamond-shaped geometric figure defined by the following vertices, - V₁ = that is positioned in the x3=2 plane. We want to rotate the diamond about its centroid e (with a positive angle) so it is positioned in the x1=e₁ plane. What is the affine map that achieves this and what are the mapped vertices v? V₂= V₂= V₁=