Discrete Mathematics with Graph Theory (Classic Version) (3rd Edition) (Pearson Modern Classics for Advanced Mathematics Series)
3rd Edition
ISBN: 9780134689555
Author: Edgar Goodaire, Michael Parmenter
Publisher: PEARSON
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Textbook Question
Chapter 9.2, Problem 24E
[BB](requires calculus) Prove that the number of edges in a bipartite graph with n vertices is at most
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Students have asked these similar questions
How long can a path be in the complete bipartite graph k(m,n)?
Give an example of a bipartite graph on 10 vertices that is
(a) 2-regular;
(b) 3-regular;
(c) 4-regular;
(d) 5-regular.
Use the formula
deg(v) = 2|E(G)| to find the number of edges of the following
vЄV (G)
graphs. Classify (count) the vertices by number of neighbors.
(a) V(G) = [100]. Edges: for all n and m in [100], n ‡ m, n is adjacent to m if and
only if |nm| ≤ 4.
(b) V(G) = [10] × [10]. Edges: for all (a, b) and (c,d) in [10] × [10], (a, b) ‡ (c,d),
(a, b) is adjacent to (c,d) if and only if a = c or b = d.
(c) V(G) = [10] × [10]. Edges: for all (a, b) and (c,d) in [10] × [10], (a, b) ‡ (c,d),
(a, b) is adjacent to (c,d) if and only if |ac| + |bd| = 1.
(d) V (G) = [10] × [10]. Edges: for all (a, b) and (c,d) in [10] × [10],
(a, b) is adjacent to (c,d) if and only if |a - c + b-d ≤ 2.
(a, b) ‡ (c,d),
Chapter 9 Solutions
Discrete Mathematics with Graph Theory (Classic Version) (3rd Edition) (Pearson Modern Classics for Advanced Mathematics Series)
Ch. 9.1 - (Answers can be found in the back of the book.)
1....Ch. 9.1 - (Answers can be found in the back of the book.)
2....Ch. 9.1 - (Answers can be found in the back of the book.)...Ch. 9.1 - (Answers can be found in the back of the book.)...Ch. 9.1 - (Answers can be found in the back of the book.)
5....Ch. 9.1 - Prob. 6TFQCh. 9.1 - Prob. 7TFQCh. 9.1 - Prob. 8TFQCh. 9.1 - Prob. 9TFQCh. 9.1 - Prob. 10TFQ
Ch. 9.1 - 1. [BB](Fictitious) A recently discovered map of...Ch. 9.1 - Prob. 2ECh. 9.1 - 3. One of the owners of the houses in the Three...Ch. 9.1 - Prob. 4ECh. 9.1 - Prob. 5ECh. 9.1 - Prob. 6ECh. 9.1 - You and a friend meet three other couples at a...Ch. 9.1 - 8. (a) A graph has six vertices, every two of...Ch. 9.1 - [BB] A graph has six vertices, every two of which...Ch. 9.1 - Prob. 10ECh. 9.1 - Prob. 11ECh. 9.2 - Prob. 1TFQCh. 9.2 - Prob. 2TFQCh. 9.2 - Prob. 3TFQCh. 9.2 - (Answers can be found in the back of the book.) is...Ch. 9.2 - Prob. 5TFQCh. 9.2 - Prob. 6TFQCh. 9.2 - Prob. 7TFQCh. 9.2 - Prob. 8TFQCh. 9.2 - Prob. 9TFQCh. 9.2 - Prob. 10TFQCh. 9.2 - Prob. 1ECh. 9.2 - Prob. 2ECh. 9.2 - Prob. 3ECh. 9.2 - Prob. 4ECh. 9.2 - Prob. 5ECh. 9.2 - Prob. 6ECh. 9.2 - Prob. 7ECh. 9.2 - Draw a graph with 64 vertices representing the...Ch. 9.2 - Consider again the graph accompanying Exercise 5...Ch. 9.2 - Prob. 10ECh. 9.2 - Prob. 11ECh. 9.2 - Prob. 12ECh. 9.2 - 13. [BB] At most social functions, there is a lot...Ch. 9.2 - Prob. 14ECh. 9.2 - 15. [BB;(a)] for each pair of graphs shown,...Ch. 9.2 - Prob. 16ECh. 9.2 - Prob. 17ECh. 9.2 - For each of the following sequences, determine if...Ch. 9.2 - Prob. 19ECh. 9.2 - [BB] A graph has five vertices of degree 4 and two...Ch. 9.2 - Determine whether each of the graphs in Fig 9.23...Ch. 9.2 - Prob. 22ECh. 9.2 - Prob. 23ECh. 9.2 - 24. [BB](requires calculus) Prove that the number...Ch. 9.2 - Prob. 25ECh. 9.2 - Prob. 26ECh. 9.2 - Prob. 27ECh. 9.2 - Prob. 28ECh. 9.2 - Prob. 29ECh. 9.2 - Prob. 30ECh. 9.2 - Prob. 31ECh. 9.2 - Prob. 32ECh. 9.2 - Prob. 33ECh. 9.2 - Prob. 34ECh. 9.2 - Prob. 35ECh. 9.3 - (Answers can be found in the back of the book.) It...Ch. 9.3 - Prob. 2TFQCh. 9.3 - Prob. 3TFQCh. 9.3 - Prob. 4TFQCh. 9.3 - Prob. 5TFQCh. 9.3 - (Answers can be found in the back of the book.)
6....Ch. 9.3 - (Answers can be found in the back of the book.) If...Ch. 9.3 - Prob. 8TFQCh. 9.3 - Prob. 9TFQCh. 9.3 - Prob. 10TFQCh. 9.3 - [BB] For each of the ten pairs of graphs that can...Ch. 9.3 - Prob. 2ECh. 9.3 - [BB] Draw all nonisomorphic graphs on n =3...Ch. 9.3 - [BB;(b)] for each pair of grpahs shown. If the...Ch. 9.3 - Prob. 5ECh. 9.3 - Prob. 6ECh. 9.3 - Prob. 7ECh. 9.3 - [BB] Prove that two graphs that are isomorphic...Ch. 9.3 - Consider the following three graphs. [BB] How many...Ch. 9.3 - Prob. 10ECh. 9.3 - Prob. 11ECh. 9 - 1. In the Konigsberg Bridge Problem, a tragic fire...Ch. 9 - 2. (a) Draw a configuration of four houses and two...Ch. 9 - 3. Find the solutions, where possible, for the...Ch. 9 - Draw a graph with six vertices at least three of...Ch. 9 - For each of the following sequences, determine if...Ch. 9 - 6. (a) Does there exist a graph with degree...Ch. 9 - Determine whether or not each of the following...Ch. 9 - Answer these questions for each sequence: Does...Ch. 9 - Find a necessary and sufficient condition for the...Ch. 9 - Prob. 10RECh. 9 - Suppose a graph has 49 vertices, each of degree 4...Ch. 9 - Prob. 12RECh. 9 - A graph G has 50 edges, four vertices of degree 2,...Ch. 9 - Prob. 14RECh. 9 - For each pair of graphs shown in fig 9.30 If the...Ch. 9 - Prob. 16RECh. 9 - 17. For each of the following cases, explain why...Ch. 9 - George is examining three graphs G1, G2, G3. He...Ch. 9 - Answer Exercise 18 again, assuming that Georges...Ch. 9 - Prob. 20RE
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- Using the table from Exercise 5, sketch the graph of 2x+3y=12.arrow_forwardGraph: 3x+2y6.arrow_forwardGrid Grove is a neighborhood, with houses organized in m rows of n columns. Houses that are closest to each other are connected by a path (note that this organization follows the definition of a grid graph given in lecture). Assume that m, n > 2. As follows from lecture, Grid Grove has mn houses and 2mn – m -n paths. It is also possible to walk to any house from any other house through some sequence of paths. To save money, the landlords want to get rid of some paths. Calculate D, the maximum number of paths that can be removed from the neighborhood without disconnecting it. Justify your answer. Then describe (informally) which D paths of the neighborhood can be removed (there is more than one such set of D paths).arrow_forward
- Count the number v of vertices, the number e of edges and the number f of regions (face) of each graph and verify Euler's formula. Also find the degree of the outside region| (b) (c) karrow_forwardB. U2 V₂ V3 D. V4 E. 05 F. 06 G. V7 H. V8 I. None of the above V₁ Which of the following vertices are adjacent to vertex v4? | Α. 01 V₁ N 6 V₁₂₁ V₂ 8arrow_forwardI want this to be considered as a Advanced Math question pls. . Consider a graph G which is a complete bipartite graph. The graph G is defined as K(3,4), meaning it has two sets of vertices, with 3 vertices in one set and 4 in the other. Every vertex in one set is connected to every vertex in the other set, but there are no connections within a set. Calculate the number of edges in graph G. Also, determine if the graph G contains an Euler path or circuit, and justify your answer.arrow_forward
- Q. 6: State and prove closed Graph theorem.arrow_forward(Discrete Math-Graph Theory) Let G=(V,E₁∪E₂) be a simple graph such that G₁=(V,E₁) and G₂=(V,E₂) are both forests. Prove or disprove: G must be a planar graph.arrow_forward2 5 10 6 5 10 11 16 9 14 10 1. Degree of a Vertex - Enumerate all vertices in this graph and identify their degrees.arrow_forward
- Quadrilateral ABCDABCD has vertices A(3,6)A(3,6), B(−4,9)B(−4,9), C(7,−5)C(7,−5), and D(1,1)D(1,1). Quadrilateral JKLMJKLM has vertices J(−2,1)J(−2,1), K(−9,4)K(−9,4), L(2,−10)L(2,−10), and M(−4,0)M(−4,0). Are the quadrilaterals congruent?arrow_forwardEuler’s formula (v − e + f = 2) holds for all connected planar graphs.What if a graph is not connected? Suppose a planar graph has twocomponents. What is the value of v − e + f now? What if it has kcomponents?arrow_forwardA quadrilateral has the vertices D (-4, -1), E (0, -4), F (6.0) and G (0.4). Verify for each pair of adjacent sides that the line segment joining their midpoints is parallel to a diagonal of the quadrilateral.arrow_forward
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