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4. Construct all degree sequences for graphs with four vertices and no isolated vertex.

4. Construct all degree sequences for graphs with four vertices and no isolated vertex.

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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MATHEMATICS IN THE MODERN WORLD 

Answer only number 4

Question 1 Name Section: 040 Do the following: 1. Find a graph with 12 edges having six vertices of degree three and the remaining vertices of degree less than three. 2. Give an example of a graph with at least four vertices, or prove that none exists, such that: (a) There are no vertices of odd degree. (b) There are no vertices of even degree. (c) There is exactly one vertex of odd degree. (d) There is exactly one vertex of even degree. (e) There are exactly two vertices of odd degree. 3. (a) Construct a graph with six vertices and degree sequence 1, 1. 2. 2. 3. 3. (b) Construct a graph with six vertices and degree sequence 1, 1, 3, 3, 3, 3. (c) Can you find at least two graphs with each of these degree sequences? 4. Construct all degree sequences for graphs with four vertices and no isolated vertex. 5. Determine all possible degree sequences for graphs with five vertices containing no isolated vertex and six edges. 6. Determine all possible degree sequences for graphs with five vertices containing no isolated vertex and eight edges. 7. Prove that the graphs G & H are isomorphic. 2 ** G 3 ●5 H
Transcribed Image Text:Question 1 Name Section: 040 Do the following: 1. Find a graph with 12 edges having six vertices of degree three and the remaining vertices of degree less than three. 2. Give an example of a graph with at least four vertices, or prove that none exists, such that: (a) There are no vertices of odd degree. (b) There are no vertices of even degree. (c) There is exactly one vertex of odd degree. (d) There is exactly one vertex of even degree. (e) There are exactly two vertices of odd degree. 3. (a) Construct a graph with six vertices and degree sequence 1, 1. 2. 2. 3. 3. (b) Construct a graph with six vertices and degree sequence 1, 1, 3, 3, 3, 3. (c) Can you find at least two graphs with each of these degree sequences? 4. Construct all degree sequences for graphs with four vertices and no isolated vertex. 5. Determine all possible degree sequences for graphs with five vertices containing no isolated vertex and six edges. 6. Determine all possible degree sequences for graphs with five vertices containing no isolated vertex and eight edges. 7. Prove that the graphs G & H are isomorphic. 2 ** G 3 ●5 H
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