V₁1 V₂ Suppose G is a undirected graph with vertices V₁, V2,V3,V4,V5 with adjacency matrix V3 V4 V5 Find (a) the number of edges in G. Answer: (b) the degree of the vertex V₂. Answer: (c) the number of loops in G. Answer: V1 V2 V3 V4 V5 01314 12001 3 0 1 03 10021 4 1 3 1 3
V₁1 V₂ Suppose G is a undirected graph with vertices V₁, V2,V3,V4,V5 with adjacency matrix V3 V4 V5 Find (a) the number of edges in G. Answer: (b) the degree of the vertex V₂. Answer: (c) the number of loops in G. Answer: V1 V2 V3 V4 V5 01314 12001 3 0 1 03 10021 4 1 3 1 3
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 25
**Suppose G is an undirected graph with vertices \( V_1, V_2, V_3, V_4, V_5 \) with the following adjacency matrix:**
\[
\begin{pmatrix}
0 & 1 & 3 & 1 & 4 \\
1 & 2 & 0 & 0 & 1 \\
3 & 0 & 1 & 0 & 3 \\
1 & 0 & 0 & 2 & 1 \\
4 & 1 & 3 & 1 & 3 \\
\end{pmatrix}
\]
**Find:**
*(a) the number of edges in G.*
*Answer: \_\_\_\_\_\_*
*(b) the degree of the vertex \( V_2 \).*
*Answer: \_\_\_\_\_\_*
*(c) the number of loops in G.*
*Answer: \_\_\_\_\_\_*
---
### Explanation of the Adjacency Matrix
The adjacency matrix provided represents an undirected graph with vertices \( V_1, V_2, V_3, V_4, V_5 \). In this matrix:
- Each element \( a_{ij} \) denotes the number of edges between the vertex \( V_i \) and vertex \( V_j \).
- The diagonal elements \( a_{ii} \) represent the number of loops at the vertex \( V_i \).
**Interpreting and Solving the Questions:**
1. **Number of edges in G:**
The sum of all the elements in the adjacency matrix divided by 2 (since each edge is represented twice, once for each vertex it connects).
2. **Degree of vertex \( V_2 \):**
The sum of the elements in the row corresponding to \( V_2 \).
3. **Number of loops in G:**
The sum of the diagonal elements in the matrix (each loop is only counted once).
By analyzing the matrix and performing the required operations, one can accurately determine the answers to the given questions about the graph's properties.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F671c99ce-18c4-47bd-a97a-569bceed506c%2Faeac864b-1b4e-4ecf-b0e5-2dc999dcf710%2Fzmv5p8w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question 25
**Suppose G is an undirected graph with vertices \( V_1, V_2, V_3, V_4, V_5 \) with the following adjacency matrix:**
\[
\begin{pmatrix}
0 & 1 & 3 & 1 & 4 \\
1 & 2 & 0 & 0 & 1 \\
3 & 0 & 1 & 0 & 3 \\
1 & 0 & 0 & 2 & 1 \\
4 & 1 & 3 & 1 & 3 \\
\end{pmatrix}
\]
**Find:**
*(a) the number of edges in G.*
*Answer: \_\_\_\_\_\_*
*(b) the degree of the vertex \( V_2 \).*
*Answer: \_\_\_\_\_\_*
*(c) the number of loops in G.*
*Answer: \_\_\_\_\_\_*
---
### Explanation of the Adjacency Matrix
The adjacency matrix provided represents an undirected graph with vertices \( V_1, V_2, V_3, V_4, V_5 \). In this matrix:
- Each element \( a_{ij} \) denotes the number of edges between the vertex \( V_i \) and vertex \( V_j \).
- The diagonal elements \( a_{ii} \) represent the number of loops at the vertex \( V_i \).
**Interpreting and Solving the Questions:**
1. **Number of edges in G:**
The sum of all the elements in the adjacency matrix divided by 2 (since each edge is represented twice, once for each vertex it connects).
2. **Degree of vertex \( V_2 \):**
The sum of the elements in the row corresponding to \( V_2 \).
3. **Number of loops in G:**
The sum of the diagonal elements in the matrix (each loop is only counted once).
By analyzing the matrix and performing the required operations, one can accurately determine the answers to the given questions about the graph's properties.
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