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Find the adjacency matrices for the directed graphs in (a) and (b). (a) V3 V2 For the adjacency matrix, the rows and columns are ordered v, through v3. 1

Find the adjacency matrices for the directed graphs in (a) and (b). (a) V3 V2 For the adjacency matrix, the rows and columns are ordered v, through v3. 1

Advanced Engineering Mathematics
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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### Graph and Adjacency Matrix (Educational Resource) #### Graph Representation The following diagram represents a directed graph with vertices and edges labeled as follows: - **Vertices (V):** - \(v_1 \) - \(v_2 \) - \(v_3 \) - \(v_4 \) - **Edges (E):** - \(e_1\) from \(v_1\) to itself (self-loop) - \(e_2\) from \(v_1\) to \(v_3\) - \(e_3\) from \(v_1\) to \(v_2\) - \(e_4\) from \(v_2\) to \(v_3\) - \(e_5\) from \(v_3\) to \(v_4\) - \(e_6\) from \(v_4\) to \(v_4\) (self-loop) #### Adjacency Matrix The adjacency matrix for the given directed graph is ordered with rows and columns corresponding to the vertices \(v_1 \) through \(v_4 \): \[ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ \end{bmatrix} \] - **Explanation of the adjacency matrix:** - The element at row \(i\) and column \(j\) is '1' if there is a directed edge from vertex \(v_i\) to vertex \(v_j\); otherwise, it is '0'. - For example, the element in the first row and the first column is '1', indicating a self-loop at \(v_1 \). - Similarly, the element in the third row and the first column is '1', showing a directed edge from \(v_3\) to \(v_1 \). #### Visual Directions: - Green arrows highlight the row and column currently being explained. - Red cross (incorrect mark) points to an error or point of attention. This detailed explanation aids in understanding graph theory and its matrix representations, which are fundamental in various fields such as computer science, network analysis
Transcribed Image Text:### Graph and Adjacency Matrix (Educational Resource) #### Graph Representation The following diagram represents a directed graph with vertices and edges labeled as follows: - **Vertices (V):** - \(v_1 \) - \(v_2 \) - \(v_3 \) - \(v_4 \) - **Edges (E):** - \(e_1\) from \(v_1\) to itself (self-loop) - \(e_2\) from \(v_1\) to \(v_3\) - \(e_3\) from \(v_1\) to \(v_2\) - \(e_4\) from \(v_2\) to \(v_3\) - \(e_5\) from \(v_3\) to \(v_4\) - \(e_6\) from \(v_4\) to \(v_4\) (self-loop) #### Adjacency Matrix The adjacency matrix for the given directed graph is ordered with rows and columns corresponding to the vertices \(v_1 \) through \(v_4 \): \[ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ \end{bmatrix} \] - **Explanation of the adjacency matrix:** - The element at row \(i\) and column \(j\) is '1' if there is a directed edge from vertex \(v_i\) to vertex \(v_j\); otherwise, it is '0'. - For example, the element in the first row and the first column is '1', indicating a self-loop at \(v_1 \). - Similarly, the element in the third row and the first column is '1', showing a directed edge from \(v_3\) to \(v_1 \). #### Visual Directions: - Green arrows highlight the row and column currently being explained. - Red cross (incorrect mark) points to an error or point of attention. This detailed explanation aids in understanding graph theory and its matrix representations, which are fundamental in various fields such as computer science, network analysis
### Understanding Adjacency Matrices for Directed Graphs **Task:** Find the adjacency matrices for the directed graphs in (a) and (b). #### (a) The directed graph contains three vertices \(v_1\), \(v_2\), and \(v_3\). It includes the following edges: - \(e_1\): From \(v_1\) to \(v_2\) - \(e_2\): From \(v_1\) to \(v_2\) - \(e_3\): From \(v_3\) to \(v_1\) Here is a visual representation of the graph: ![Graph](attachment:image.png) #### Adjacency Matrix For the adjacency matrix, the rows and columns are ordered \(v_1\) through \(v_3\): \[ \begin{bmatrix} 0 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{bmatrix} \] - The element in the first row and second column corresponds to the edge \(e_1\) (from \(v_1\) to \(v_2\)). - The element in the first row and third column corresponds to the edge \(e_2\) (from \(v_1\) to \(v_2\)). - The element in the second row and third column corresponds to the edge \(e_3\) (from \(v_3\) to \(v_1\)). Therefore, the adjacency matrix is: \[ \begin{bmatrix} 0 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{bmatrix} \] This matrix indicates the presence and direction of edges between vertices, where a value of \(1\) specifies a directed edge from the row vertex to the column vertex.
Transcribed Image Text:### Understanding Adjacency Matrices for Directed Graphs **Task:** Find the adjacency matrices for the directed graphs in (a) and (b). #### (a) The directed graph contains three vertices \(v_1\), \(v_2\), and \(v_3\). It includes the following edges: - \(e_1\): From \(v_1\) to \(v_2\) - \(e_2\): From \(v_1\) to \(v_2\) - \(e_3\): From \(v_3\) to \(v_1\) Here is a visual representation of the graph: ![Graph](attachment:image.png) #### Adjacency Matrix For the adjacency matrix, the rows and columns are ordered \(v_1\) through \(v_3\): \[ \begin{bmatrix} 0 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{bmatrix} \] - The element in the first row and second column corresponds to the edge \(e_1\) (from \(v_1\) to \(v_2\)). - The element in the first row and third column corresponds to the edge \(e_2\) (from \(v_1\) to \(v_2\)). - The element in the second row and third column corresponds to the edge \(e_3\) (from \(v_3\) to \(v_1\)). Therefore, the adjacency matrix is: \[ \begin{bmatrix} 0 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{bmatrix} \] This matrix indicates the presence and direction of edges between vertices, where a value of \(1\) specifies a directed edge from the row vertex to the column vertex.
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