Chapter 2.3, Problem 54E

a.

To determine

To represent: the graph with $A,B,C,\text{ and }D$ nodes using a matrix

Answer to Problem 54E

  $$

  $\begin{array}{cc}\begin{array}{l}\begin{array}{c}\\ A\\ B\\ C\end{array}\\ D\end{array}& \begin{array}{l}\begin{array}{cccc}A& B& C& D\end{array}\\ \left[\begin{array}{cccc}\begin{array}{l}0\\ 1\\ 0\\ 0\end{array}& \begin{array}{l}1\\ 0\\ 1\\ 1\end{array}& \begin{array}{l}0\\ 1\\ 0\\ 2\end{array}& \begin{array}{l}0\\ 1\\ 2\\ 0\end{array}\end{array}\right]\end{array}\end{array}$

Explanation of Solution

Given information:

The finite graph given is shown below.

  

The finite graph contains points called nodes and segments called edges. The graphs can be represented by the square matrices. The element of the matrix is the number of edges between each pair of nodes.

The square matrix of another finite graph shown below is given of help.

  $\begin{array}{cc}\begin{array}{l}\begin{array}{c}\\ R\\ S\\ T\end{array}\\ U\end{array}& \begin{array}{l}\begin{array}{cccc}R& S& T& U\end{array}\\ \left[\begin{array}{cccc}\begin{array}{l}0\\ 2\\ 0\\ 1\end{array}& \begin{array}{l}2\\ 0\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 0\\ 0\\ 1\end{array}& \begin{array}{l}1\\ 0\\ 1\\ 0\end{array}\end{array}\right]\end{array}\end{array}$

Calculation:

Since the element of the matrix is the number of edges between each pair of nodes. Thus, the graph can be represented in matrix form as follows:

  $\begin{array}{cc}\begin{array}{l}\begin{array}{c}\\ A\\ B\\ C\end{array}\\ D\end{array}& \begin{array}{l}\begin{array}{cccc}A& B& C& D\end{array}\\ \left[\begin{array}{cccc}\begin{array}{l}0\\ 1\\ 0\\ 0\end{array}& \begin{array}{l}1\\ 0\\ 1\\ 1\end{array}& \begin{array}{l}0\\ 1\\ 0\\ 2\end{array}& \begin{array}{l}0\\ 1\\ 2\\ 0\end{array}\end{array}\right]\end{array}\end{array}$

b.

To determine

and explain whether different graphs can be represented by same matrix.

Answer to Problem 54E

  

Explanation of Solution

Given information: The finite graph contains points called nodes and segments called edges. The graphs can be represented by the square matrices. The element of the matrix is the number of edges between each pair of nodes. The finite graph given is shown below.

  

The square matrix of given finite graph shown below is given of help.

  $\begin{array}{cc}\begin{array}{l}\begin{array}{c}\\ A\\ B\\ C\end{array}\\ D\end{array}& \begin{array}{l}\begin{array}{cccc}A& B& C& D\end{array}\\ \left[\begin{array}{cccc}\begin{array}{l}0\\ 1\\ 0\\ 0\end{array}& \begin{array}{l}1\\ 0\\ 1\\ 1\end{array}& \begin{array}{l}0\\ 1\\ 0\\ 2\end{array}& \begin{array}{l}0\\ 1\\ 2\\ 0\end{array}\end{array}\right]\end{array}\end{array}$

Calculation:

Since the element of the matrix is the number of edges between each pair of nodes. Thus, the different graph can be represented for same matrix as follows: