A graph with vertices v1, V2, V3, V4, U5 has adjacency matrix 1 0 2 A =| 0 1 1 1 0 2 1. How many edges does the graph have? answer = 2. What is the degree of vertex v2? degree = 3. What is the degree of vertex v4? degree
A graph with vertices v1, V2, V3, V4, U5 has adjacency matrix 1 0 2 A =| 0 1 1 1 0 2 1. How many edges does the graph have? answer = 2. What is the degree of vertex v2? degree = 3. What is the degree of vertex v4? degree
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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![**Title:** Understanding Graphs through Adjacency Matrices
**Introduction:**
A graph with vertices \(v_1, v_2, v_3, v_4, v_5\) is represented by its adjacency matrix \(A\). The adjacency matrix is a square matrix used to describe a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.
**Adjacency Matrix:**
\[
A = \begin{bmatrix}
0 & 0 & 0 & 1 & 0 \\
0 & 2 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
1 & 0 & 1 & 0 & 2 \\
0 & 0 & 0 & 2 & 0 \\
\end{bmatrix}
\]
**Explanation of the Matrix:**
- Each row and column represent a vertex in the graph.
- A non-zero entry in the matrix \(a_{ij}\) indicates an edge between vertex \(v_i\) and vertex \(v_j\).
- The value of the entry represents the number of edges connecting the vertices if it is greater than one.
**Questions:**
1. **How many edges does the graph have?**
- **Answer:** The sum of all entries above the main diagonal in the symmetric matrix gives the total number of edges. Here, the answer is already provided as 5.
2. **What is the degree of vertex \(v_2\)?**
- **Degree:** The degree is the total number of edges connected to the vertex. Calculate the degree by summing the entries in the corresponding row or column for \(v_2\).
3. **What is the degree of vertex \(v_4\)?**
- **Degree:** Again, sum the entries in the row or column for \(v_4\) to find its degree.
**Conclusion:**
Adjacency matrices are a powerful tool for representing graphs and understanding the relationships between vertices. By interpreting these matrices, you can answer important questions about the structure and properties of a graph.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F31330aa5-a64b-4d2f-80e6-6ea74abb2a34%2F0dde67dd-e844-4ca4-885a-a9f2d0bd3668%2Fvn5raur4_processed.png&w=3840&q=75)
Transcribed Image Text:**Title:** Understanding Graphs through Adjacency Matrices
**Introduction:**
A graph with vertices \(v_1, v_2, v_3, v_4, v_5\) is represented by its adjacency matrix \(A\). The adjacency matrix is a square matrix used to describe a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.
**Adjacency Matrix:**
\[
A = \begin{bmatrix}
0 & 0 & 0 & 1 & 0 \\
0 & 2 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
1 & 0 & 1 & 0 & 2 \\
0 & 0 & 0 & 2 & 0 \\
\end{bmatrix}
\]
**Explanation of the Matrix:**
- Each row and column represent a vertex in the graph.
- A non-zero entry in the matrix \(a_{ij}\) indicates an edge between vertex \(v_i\) and vertex \(v_j\).
- The value of the entry represents the number of edges connecting the vertices if it is greater than one.
**Questions:**
1. **How many edges does the graph have?**
- **Answer:** The sum of all entries above the main diagonal in the symmetric matrix gives the total number of edges. Here, the answer is already provided as 5.
2. **What is the degree of vertex \(v_2\)?**
- **Degree:** The degree is the total number of edges connected to the vertex. Calculate the degree by summing the entries in the corresponding row or column for \(v_2\).
3. **What is the degree of vertex \(v_4\)?**
- **Degree:** Again, sum the entries in the row or column for \(v_4\) to find its degree.
**Conclusion:**
Adjacency matrices are a powerful tool for representing graphs and understanding the relationships between vertices. By interpreting these matrices, you can answer important questions about the structure and properties of a graph.
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