3. ..) Determine if the following simple graphs exist. If so, draw such a graph. If not, explain why it does not exist. (a) A simple graph with 5 vertices of degree 1,2,2,3,4. (b) A simple graph with 7 vertices of degree 1,1,2,2,3,5,5. (c) A simple graph with 6 vertices of degree 1,1,1,1,2,2.

3. ..) Determine if the following simple graphs exist. If so, draw such a graph. If not, explain why it does not exist. (a) A simple graph with 5 vertices of degree 1,2,2,3,4. (b) A simple graph with 7 vertices of degree 1,1,2,2,3,5,5. (c) A simple graph with 6 vertices of degree 1,1,1,1,2,2.

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

### Graph Theory Problem Set

#### 3. Determine if the following simple graphs exist. If so, draw such a graph. If not, explain why it does not exist.

**(a) A simple graph with 5 vertices of degree 1, 2, 2, 3, 4.**

**Answer:**
- To check if a simple graph exists with these degrees, we need to ensure that the degree sum is even (Handshaking Lemma).
- Sum of degrees: 1 + 2 + 2 + 3 + 4 = 12 (even).
- Therefore, it is possible to construct a graph with these vertex degrees.

**(b) A simple graph with 7 vertices of degree 1, 1, 2, 2, 3, 5, 5.**

**Answer:**
- Sum of degrees: 1 + 1 + 2 + 2 + 3 + 5 + 5 = 19 (odd).
- Since the degree sum is odd, it is impossible to construct a simple graph with these vertex degrees.

**(c) A simple graph with 6 vertices of degree 1, 1, 1, 1, 2, 2.**

**Answer:**
- Sum of degrees: 1 + 1 + 1 + 1 + 2 + 2 = 8 (even).
- Therefore, it is possible to construct a graph with these vertex degrees.

**(d) A simple graph on 5 vertices with 12 edges.**

**Answer:**
- For a simple graph, the maximum possible number of edges is given by the formula n(n-1)/2 where n is the number of vertices.
- For 5 vertices: 5(5-1)/2 = 10 edges.
- Since we cannot exceed 10 edges with 5 vertices, a simple graph with 12 edges on 5 vertices is impossible.

**(e) A simple graph with degrees 1, 1, 1, 1, 2, 2, 3.**

**Answer:**
- Sum of degrees: 1 + 1 + 1 + 1 + 2 + 2 + 3 = 11 (odd).
- Since the degree sum is odd, it is impossible to construct a simple graph with these vertex degrees.

**(f) A complete

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