### Graph Theory Problem - Adjacency and Incidence #### Problem Statement: Consider the graph in **Figure 1**. Which of the following statements hold true for the graph?  #### Figure Description: In Figure 1, the graph consists of: - **Vertices**: \( v_1, v_2, v_3, v_4 \) - **Edges**: \( e_1, e_2, e_3, e_4, e_5, e_6, e_7 \) The edges are connected as follows: - Edge \( e_1 \) connects vertex \( v_1 \) to vertex \( v_3 \). - Edge \( e_2 \) connects vertex \( v_3 \) to vertex \( v_4 \). - Edge \( e_3 \) connects vertex \( v_2 \) to vertex \( v_4 \). - Edge \( e_4 \) connects vertex \( v_1 \) to vertex \( v_2 \). - Edge \( e_5 \) connects vertex \( v_1 \) to vertex \( v_4 \). - Edge \( e_6 \) connects vertex \( v_3 \) to vertex \( v_4 \). - Edge \( e_7 \) forms a loop at vertex \( v_2 \). #### Statements to Evaluate: (a) Vertices \( v_3 \) and \( v_2 \) are adjacent (b) Edge \( e_6 \) is incident with vertex \( v_4 \) (c) Vertex \( v_2 \) is incident with edge \( e_4 \) (d) Vertex \( v_4 \) and edges \( e_5 \) and \( e_6 \) form a subgraph of the graph **Figure 1: Graph for Problem 1** ### Analysis: 1. **Adjacency**: Vertices \( v_3 \) and \( v_2 \) are not adjacent; there is no direct edge connecting them. 2. **Incidence with \( v_4 \)**: Edge \( e_6 \) connects vertex \( v_1 \) to vertex \( v_4 \), making it incident with \( v_4 \). 3. **Incidence with \( v_2 \)**: Edge \( e_4 \) connects vertex \( v_1
### Graph Theory Problem - Adjacency and Incidence #### Problem Statement: Consider the graph in **Figure 1**. Which of the following statements hold true for the graph?  #### Figure Description: In Figure 1, the graph consists of: - **Vertices**: \( v_1, v_2, v_3, v_4 \) - **Edges**: \( e_1, e_2, e_3, e_4, e_5, e_6, e_7 \) The edges are connected as follows: - Edge \( e_1 \) connects vertex \( v_1 \) to vertex \( v_3 \). - Edge \( e_2 \) connects vertex \( v_3 \) to vertex \( v_4 \). - Edge \( e_3 \) connects vertex \( v_2 \) to vertex \( v_4 \). - Edge \( e_4 \) connects vertex \( v_1 \) to vertex \( v_2 \). - Edge \( e_5 \) connects vertex \( v_1 \) to vertex \( v_4 \). - Edge \( e_6 \) connects vertex \( v_3 \) to vertex \( v_4 \). - Edge \( e_7 \) forms a loop at vertex \( v_2 \). #### Statements to Evaluate: (a) Vertices \( v_3 \) and \( v_2 \) are adjacent (b) Edge \( e_6 \) is incident with vertex \( v_4 \) (c) Vertex \( v_2 \) is incident with edge \( e_4 \) (d) Vertex \( v_4 \) and edges \( e_5 \) and \( e_6 \) form a subgraph of the graph **Figure 1: Graph for Problem 1** ### Analysis: 1. **Adjacency**: Vertices \( v_3 \) and \( v_2 \) are not adjacent; there is no direct edge connecting them. 2. **Incidence with \( v_4 \)**: Edge \( e_6 \) connects vertex \( v_1 \) to vertex \( v_4 \), making it incident with \( v_4 \). 3. **Incidence with \( v_2 \)**: Edge \( e_4 \) connects vertex \( v_1
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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Please do b, c and d! Please show steps and explain!
#### Figure Description:
In Figure 1, the graph consists of:
- **Vertices**: \( v_1, v_2, v_3, v_4 \)
- **Edges**: \( e_1, e_2, e_3, e_4, e_5, e_6, e_7 \)
The edges are connected as follows:
- Edge \( e_1 \) connects vertex \( v_1 \) to vertex \( v_3 \).
- Edge \( e_2 \) connects vertex \( v_3 \) to vertex \( v_4 \).
- Edge \( e_3 \) connects vertex \( v_2 \) to vertex \( v_4 \).
- Edge \( e_4 \) connects vertex \( v_1 \) to vertex \( v_2 \).
- Edge \( e_5 \) connects vertex \( v_1 \) to vertex \( v_4 \).
- Edge \( e_6 \) connects vertex \( v_3 \) to vertex \( v_4 \).
- Edge \( e_7 \) forms a loop at vertex \( v_2 \).
#### Statements to Evaluate:
(a) Vertices \( v_3 \) and \( v_2 \) are adjacent
(b) Edge \( e_6 \) is incident with vertex \( v_4 \)
(c) Vertex \( v_2 \) is incident with edge \( e_4 \)
(d) Vertex \( v_4 \) and edges \( e_5 \) and \( e_6 \) form a subgraph of the graph
**Figure 1: Graph for Problem 1**
### Analysis:
1. **Adjacency**: Vertices \( v_3 \) and \( v_2 \) are not adjacent; there is no direct edge connecting them.
2. **Incidence with \( v_4 \)**: Edge \( e_6 \) connects vertex \( v_1 \) to vertex \( v_4 \), making it incident with \( v_4 \).
3. **Incidence with \( v_2 \)**: Edge \( e_4 \) connects vertex \( v_1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc899dcba-268c-4add-8443-07635494adc0%2F40eff9e7-f687-44c9-837b-bc3c4bf8c892%2Fl632uc.jpeg&w=3840&q=75)
Transcribed Image Text:### Graph Theory Problem - Adjacency and Incidence
#### Problem Statement:
Consider the graph in **Figure 1**. Which of the following statements hold true for the graph?
#### Figure Description:
In Figure 1, the graph consists of:
- **Vertices**: \( v_1, v_2, v_3, v_4 \)
- **Edges**: \( e_1, e_2, e_3, e_4, e_5, e_6, e_7 \)
The edges are connected as follows:
- Edge \( e_1 \) connects vertex \( v_1 \) to vertex \( v_3 \).
- Edge \( e_2 \) connects vertex \( v_3 \) to vertex \( v_4 \).
- Edge \( e_3 \) connects vertex \( v_2 \) to vertex \( v_4 \).
- Edge \( e_4 \) connects vertex \( v_1 \) to vertex \( v_2 \).
- Edge \( e_5 \) connects vertex \( v_1 \) to vertex \( v_4 \).
- Edge \( e_6 \) connects vertex \( v_3 \) to vertex \( v_4 \).
- Edge \( e_7 \) forms a loop at vertex \( v_2 \).
#### Statements to Evaluate:
(a) Vertices \( v_3 \) and \( v_2 \) are adjacent
(b) Edge \( e_6 \) is incident with vertex \( v_4 \)
(c) Vertex \( v_2 \) is incident with edge \( e_4 \)
(d) Vertex \( v_4 \) and edges \( e_5 \) and \( e_6 \) form a subgraph of the graph
**Figure 1: Graph for Problem 1**
### Analysis:
1. **Adjacency**: Vertices \( v_3 \) and \( v_2 \) are not adjacent; there is no direct edge connecting them.
2. **Incidence with \( v_4 \)**: Edge \( e_6 \) connects vertex \( v_1 \) to vertex \( v_4 \), making it incident with \( v_4 \).
3. **Incidence with \( v_2 \)**: Edge \( e_4 \) connects vertex \( v_1
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