Question

consider the given network. represent the network in a matrix form.

\\( a = \

$$\begin{bmatrix} 0 & 1 & 0 \\\\ 1 & 1 & 1 \\\\ 0 & 1 & 0 \\end{bmatrix}$$

\\)

\\( a = \

$$\begin{bmatrix} 0 & 1 & 0 \\\\ 1 & 0 & 1 \\\\ 0 & 1 & 0 \\end{bmatrix}$$

\\)

Explanation:

Identify network vertices and connections

The network consists of three nodes: \(X\), \(Y\), and \(Z\).
We analyze the bidirectional connections (edges) between them:

• There is a bidirectional connection between \(X\) and \(Y\).
• There is a bidirectional connection between \(Y\) and \(Z\).
• There is no connection between \(X\) and \(Z\).
• There are no self-loops (connections from a node to itself).

Construct the adjacency matrix

Using the Matrix Representation concept, we set up a \(3 \times 3\) matrix \(A\) where rows and columns correspond to nodes \(X\), \(Y\), and \(Z\) in alphabetical order:

• Row 1 (\(X\)): Connects to \(Y\). Elements: \([0, 1, 0]\)
• Row 2 (\(Y\)): Connects to \(X\) and \(Z\). Elements: \([1, 0, 1]\)
• Row 3 (\(Z\)): Connects to \(Y\). Elements: \([0, 1, 0]\)

This yields the matrix:

$$ A =
LATEXBLOCK0
$$

Match with the given options

We compare our constructed matrix with the provided choices:

• Option 1: \(A =

$$\begin{bmatrix} 0 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 0 \end{bmatrix}$$

\) (Incorrect, implies a self-loop at \(Y\))

• Option 2: \(A =

$$\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$$

\) (Correct)

Answer:

• (A) \(A =

$$\begin{bmatrix} 0 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 0 \end{bmatrix}$$

\)

• **(B) \(A =

$$\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$$

\) (Correct answer)**