Finite Mathematics for Business, Economics, Life Sciences and Social Sciences
14th Edition
ISBN: 9780134677972
Author: Barnett
Publisher: PEARSON
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Chapter DPT, Problem 36E
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7. [10 marks]
Let G
=
(V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a
cycle in G on which x, y, and z all lie.
(a) First prove that there are two internally disjoint xy-paths Po and P₁.
(b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which
x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that
there are three paths Qo, Q1, and Q2 such that:
⚫each Qi starts at z;
• each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are
distinct;
the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex
2) and are disjoint from the paths Po and P₁ (except at the end vertices wo,
W1, and w₂).
(c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and
z all lie. (To do this, notice that two of the w; must be on the same Pj.)
6. [10 marks]
Let T be a tree with n ≥ 2 vertices and leaves. Let BL(T) denote the block graph of
T.
(a) How many vertices does BL(T) have?
(b) How many edges does BL(T) have?
Prove that your answers are correct.
4. [10 marks]
Find both a matching of maximum size and a vertex cover of minimum size in
the following bipartite graph. Prove that your answer is correct.
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Chapter DPT Solutions
Finite Mathematics for Business, Economics, Life Sciences and Social Sciences
Ch. DPT - Replace each question mark with an appropriate...Ch. DPT - Problems 2-6 refer to the following polynomials:...Ch. DPT - Problems 2-6 refer to the following polynomials:...Ch. DPT - Problems 2-6 refer to the following polynomials:...Ch. DPT - Problems 2-6 refer to the following polynomials:...Ch. DPT - Problems 2-6 refer to the following polynomials:...Ch. DPT - In Problems 7 and 8, perform the indicated...Ch. DPT - In Problems 7 and 8, perform the indicated...Ch. DPT - In Problems 9 and 10, factor completely. x2+7x+10Ch. DPT - In Problems 9 and 10, factor completely. x32x215x
Ch. DPT - Write 0.35 as a fraction reduced to lowest terms.Ch. DPT - Write 78 in decimal form.Ch. DPT - Write in scientific notation:...Ch. DPT - Write in standard decimal form: A2.55108B4.06104Ch. DPT - Indicate true (T) or false (F): (A) A natural...Ch. DPT - Give an example of an integer that is not a...Ch. DPT - In Problems 17-24, simplify and write answers...Ch. DPT - In Problems 17-24, simplify and write answers...Ch. DPT - In Problems 17-24, simplify and write answers...Ch. DPT - In Problems 17-24, simplify and write answers...Ch. DPT - In Problems 17-24, simplify and write answers...Ch. DPT - In Problems 17-24, simplify and write answers...Ch. DPT - In Problems 17-24, simplify and write answers...Ch. DPT - In Problems 17-24, simplify and write answers...Ch. DPT - In Problems 25-30, perform the indicated operation...Ch. DPT - In Problems 25-30, perform the indicated operation...Ch. DPT - In Problems 25-30, perform the indicated operation...Ch. DPT - In Problems 25-30, perform the indicated operation...Ch. DPT - In Problems 25-30, perform the indicated operation...Ch. DPT - In Problems 25-30, perform the indicated operation...Ch. DPT - Each statement illustrates the use of one of the...Ch. DPT - Prob. 32ECh. DPT - Multiplying a number x by 4 gives the same result...Ch. DPT - Find the slope of the line that contains the...Ch. DPT - Find the x and y coordinates of the point at which...Ch. DPT - Find the x and y coordinates of the point at which...Ch. DPT - In Problems 37-40, solve for x. x2=5xCh. DPT - In Problems 37-40, solve for x. 3x221=0Ch. DPT - In Problems 37-40, solve for x. x2x20=0Ch. DPT - In Problems 37-40, solve for x. 6x2+7x1=0
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- 5. [10 marks] Let G = (V,E) be a graph, and let X C V be a set of vertices. Prove that if |S||N(S)\X for every SCX, then G contains a matching M that matches every vertex of X (i.e., such that every x X is an end of an edge in M).arrow_forwardQ/show that 2" +4 has a removable discontinuity at Z=2i Z(≥2-21)arrow_forwardRefer to page 100 for problems on graph theory and linear algebra. Instructions: • Analyze the adjacency matrix of a given graph to find its eigenvalues and eigenvectors. • Interpret the eigenvalues in the context of graph properties like connectivity or clustering. Discuss applications of spectral graph theory in network analysis. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440 AZF/view?usp=sharing]arrow_forward
- Refer to page 110 for problems on optimization. Instructions: Given a loss function, analyze its critical points to identify minima and maxima. • Discuss the role of gradient descent in finding the optimal solution. . Compare convex and non-convex functions and their implications for optimization. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 140 for problems on infinite sets. Instructions: • Compare the cardinalities of given sets and classify them as finite, countable, or uncountable. • Prove or disprove the equivalence of two sets using bijections. • Discuss the implications of Cantor's theorem on real-world computation. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 120 for problems on numerical computation. Instructions: • Analyze the sources of error in a given numerical method (e.g., round-off, truncation). • Compute the error bounds for approximating the solution of an equation. • Discuss strategies to minimize error in iterative methods like Newton-Raphson. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forward
- Refer to page 145 for problems on constrained optimization. Instructions: • Solve an optimization problem with constraints using the method of Lagrange multipliers. • • Interpret the significance of the Lagrange multipliers in the given context. Discuss the applications of this method in machine learning or operations research. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forwardOnly 100% sure experts solve it correct complete solutions okarrow_forwardGive an example of a graph with at least 3 vertices that has exactly 2 automorphisms(one of which is necessarily the identity automorphism). Prove that your example iscorrect.arrow_forward
- 3. [10 marks] Let Go (Vo, Eo) and G₁ = (V1, E1) be two graphs that ⚫ have at least 2 vertices each, ⚫are disjoint (i.e., Von V₁ = 0), ⚫ and are both Eulerian. Consider connecting Go and G₁ by adding a set of new edges F, where each new edge has one end in Vo and the other end in V₁. (a) Is it possible to add a set of edges F of the form (x, y) with x € Vo and y = V₁ so that the resulting graph (VUV₁, Eo UE₁ UF) is Eulerian? (b) If so, what is the size of the smallest possible F? Prove that your answers are correct.arrow_forwardLet T be a tree. Prove that if T has a vertex of degree k, then T has at least k leaves.arrow_forwardHomework Let X1, X2, Xn be a random sample from f(x;0) where f(x; 0) = (-), 0 < x < ∞,0 € R Using Basu's theorem, show that Y = min{X} and Z =Σ(XY) are indep. -arrow_forward
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