Find the adjacency matrices for the directed graphs in (a) and (b). **Finding Adjacency Matrices for Directed Graphs****Graph (a):**- **Vertices and Edges:** - The graph consists of three vertices: $ v_1, v_2, v_3 $. - Edges are directed as follows: - $ e_1 $ from $ v_1 $ to $ v_2 $ - $ e_2 $ from $ v_2 $ to $ v_1 $ - $ e_3 $ from $ v_3 $ to $ v_1 $- **Adjacency Matrix:** - The rows and columns of the matrix are ordered $ v_1, v_2, v_3 $. - The matrix is: \[ \begin{bmatrix} 0 & 2 & 1 \\ 2 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix} \] - Explanation: - The entry at row $ i $, column $ j $ represents the number of directed edges from vertex $ v_i $ to vertex $ v_j $. - For example, there are two edges from $ v_1 $ to $ v_2 $ as indicated by the value 2 in the first row, second column.**Graph (b):**- **Vertices and Edges:** - The graph consists of four vertices: $ v_1, v_2, v_3, v_4 $. - Edges are directed as follows: - $ e_1 $ is a loop from $ v_1 $ to $ v_1 $ - $ e_2 $ from $ v_1 $ to $ v_3 $ - $ e_3 $ from $ v_3 $ to $ v_1 $ - $ e_4 $ from $ v_3 $ to $ v_2 $ - $ e_5 $ from $ v_3 $ to $ v_4 $- **Adjacency Matrix:** - The rows and columns of the matrix are ordered $ v_1, v_2, v_3, v_4 $. - The matrix is: \[

Rematch: The given graphs in (a) and (b) are not diagraphs (because no edges are directed with…

Answered: Find the adjacency matrices for the…

Find the adjacency matrices for the directed graphs in (a) and (b). (a) V3 V1 V2 For the adjacency matrix, the rows and columns are ordered v, through v 2 1 2 1 (b) V2 V1

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

Find the adjacency matrices for the directed graphs in (a) and (b).

**Finding Adjacency Matrices for Directed Graphs**

**Graph (a):**

- **Vertices and Edges:**
- The graph consists of three vertices: $ v_1, v_2, v_3 $.
- Edges are directed as follows:
- $ e_1 $ from $ v_1 $ to $ v_2 $
- $ e_2 $ from $ v_2 $ to $ v_1 $
- $ e_3 $ from $ v_3 $ to $ v_1 $

- **Adjacency Matrix:**
- The rows and columns of the matrix are ordered $ v_1, v_2, v_3 $.
- The matrix is:

\[
\begin{bmatrix}
0 & 2 & 1 \\
2 & 0 & 0 \\
1 & 0 & 0
\end{bmatrix}
\]

- Explanation:
- The entry at row $ i $, column $ j $ represents the number of directed edges from vertex $ v_i $ to vertex $ v_j $.
- For example, there are two edges from $ v_1 $ to $ v_2 $ as indicated by the value 2 in the first row, second column.

**Graph (b):**

- **Vertices and Edges:**
- The graph consists of four vertices: $ v_1, v_2, v_3, v_4 $.
- Edges are directed as follows:
- $ e_1 $ is a loop from $ v_1 $ to $ v_1 $
- $ e_2 $ from $ v_1 $ to $ v_3 $
- $ e_3 $ from $ v_3 $ to $ v_1 $
- $ e_4 $ from $ v_3 $ to $ v_2 $
- $ e_5 $ from $ v_3 $ to $ v_4 $

- **Adjacency Matrix:**
- The rows and columns of the matrix are ordered $ v_1, v_2, v_3, v_4 $.
- The matrix is:

\[

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