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Question 4 Show that every u v walk in a graph contains au- v path. Question 5 [5.1] Prove or disprove that a graph and its complement cannot both be disconnected.. [5.2] Prove or disprove that if G is a connected graph, then its complement G is disconnected. Question 6 Consider the following graph G a b C d V}] e g h Determine the following (justify all your answers): 1. The order of G. 2. The size of G. 3. Two adjacent vertices in G. 4. Two nonadjacent vertices in G. 5. The open neighborhood of d. 6. The closed neighborhood of d. 7. The maximum degree of G. 8. The minimum degree of G. 9. The degree sequence of G. Question 7 During the Covid-19 lockdown, a group of 10 people met around a diner party. They each shook hands with each other, how many handshakes took place in that diner? Hint: Use vertices and edges. END Question 2 [2.1] Suppose that we know the degrees of the vertices of a graph G. Is it possible to determine the order and size of G? Explain. [2.2] Let G be a graph of order p and size q in which every vertex has degree k, where k is odd. Prove that q is a multiple of k. [2.3] Show that, in every graph, the number of vertices having odd degree is even.

Question 4 Show that every u v walk in a graph contains au- v path. Question 5 [5.1] Prove or disprove that a graph and its complement cannot both be disconnected.. [5.2] Prove or disprove that if G is a connected graph, then its complement G is disconnected. Question 6 Consider the following graph G a b C d V}] e g h Determine the following (justify all your answers): 1. The order of G. 2. The size of G. 3. Two adjacent vertices in G. 4. Two nonadjacent vertices in G. 5. The open neighborhood of d. 6. The closed neighborhood of d. 7. The maximum degree of G. 8. The minimum degree of G. 9. The degree sequence of G. Question 7 During the Covid-19 lockdown, a group of 10 people met around a diner party. They each shook hands with each other, how many handshakes took place in that diner? Hint: Use vertices and edges. END Question 2 [2.1] Suppose that we know the degrees of the vertices of a graph G. Is it possible to determine the order and size of G? Explain. [2.2] Let G be a graph of order p and size q in which every vertex has degree k, where k is odd. Prove that q is a multiple of k. [2.3] Show that, in every graph, the number of vertices having odd degree is even.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 4 Show that every u v walk in a graph contains au- v path. Question 5 [5.1] Prove or disprove that a graph and its complement cannot both be disconnected.. [5.2] Prove or disprove that if G is a connected graph, then its complement G is disconnected. Question 6 Consider the following graph G a b C d V}] e g h Determine the following (justify all your answers): 1. The order of G. 2. The size of G. 3. Two adjacent vertices in G. 4. Two nonadjacent vertices in G. 5. The open neighborhood of d. 6. The closed neighborhood of d. 7. The maximum degree of G. 8. The minimum degree of G. 9. The degree sequence of G. Question 7 During the Covid-19 lockdown, a group of 10 people met around a diner party. They each shook hands with each other, how many handshakes took place in that diner? Hint: Use vertices and edges. END
Transcribed Image Text:Question 4 Show that every u v walk in a graph contains au- v path. Question 5 [5.1] Prove or disprove that a graph and its complement cannot both be disconnected.. [5.2] Prove or disprove that if G is a connected graph, then its complement G is disconnected. Question 6 Consider the following graph G a b C d V}] e g h Determine the following (justify all your answers): 1. The order of G. 2. The size of G. 3. Two adjacent vertices in G. 4. Two nonadjacent vertices in G. 5. The open neighborhood of d. 6. The closed neighborhood of d. 7. The maximum degree of G. 8. The minimum degree of G. 9. The degree sequence of G. Question 7 During the Covid-19 lockdown, a group of 10 people met around a diner party. They each shook hands with each other, how many handshakes took place in that diner? Hint: Use vertices and edges. END
Question 2 [2.1] Suppose that we know the degrees of the vertices of a graph G. Is it possible to determine the order and size of G? Explain. [2.2] Let G be a graph of order p and size q in which every vertex has degree k, where k is odd. Prove that q is a multiple of k. [2.3] Show that, in every graph, the number of vertices having odd degree is even.
Transcribed Image Text:Question 2 [2.1] Suppose that we know the degrees of the vertices of a graph G. Is it possible to determine the order and size of G? Explain. [2.2] Let G be a graph of order p and size q in which every vertex has degree k, where k is odd. Prove that q is a multiple of k. [2.3] Show that, in every graph, the number of vertices having odd degree is even.
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