Chapter 12.2, Problem 9E

Solution

To graph: The information in the matrix as a directed graph.

  $\begin{array}{l}\begin{array}{cccc}{N}_1& N_2& N_3& N_4\end{array}\\ \begin{array}{c}{N}_1\\ N_2\\ N_3\\ N_4\end{array}\left[\begin{array}{cccc}1& 1& 1& 1\\ 0& 0& 1& 1\\ 1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right]\end{array}$

Given information:

The given matrix is:

  $\begin{array}{l}\begin{array}{cccc}{N}_1& N_2& N_3& N_4\end{array}\\ \begin{array}{c}{N}_1\\ N_2\\ N_3\\ N_4\end{array}\left[\begin{array}{cccc}1& 1& 1& 1\\ 0& 0& 1& 1\\ 1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right]\end{array}$

Graph:

Consider the given matrix:

  $\begin{array}{l}\begin{array}{cccc}{N}_1& N_2& N_3& N_4\end{array}\\ \begin{array}{c}{N}_1\\ N_2\\ N_3\\ N_4\end{array}\left[\begin{array}{cccc}1& 1& 1& 1\\ 0& 0& 1& 1\\ 1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right]\end{array}$

A finite graph is a set of points, called vertices, connected by paths or curves.

The digit zero indicates that no path exists between two vertices or from one vertex itself.

The digit “1” indicates that a path between two vertices or one vertex and itself.

Directed graphs are finite graphs that indicate the direction of a path.

Draw the directed graph which represents to the above matrix:

  

Interpretation: The path between $N_1\text{ to }{N}_3$ is two-way. The path between $N_1\text{ to }{N}_2,N_2\text{ to }{N}_3,N_4\text{ to }{N}_3\text{, }{N}_2\text{ to }{N}_4\text{ and }{N}_1\text{ to }{N}_4$ is one way. There is a loop from $N_1$ to itself.