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9.03 Degree and adjacency

Worksheet
Degrees and adjacency matrices
1

Consider the following networks:

a
i

State the degree of vertex C.

ii

State the degree of vertex D.

b
i

State the degree of vertex X.

ii

State the degree of vertex C.

2

Create an adjacency matrix for the following network:

3

For each of the following networks:

i

Create an adjacency matrix for the network.

ii

State the degree of each of the vertex.

a
b
c
d
4

Consider this adjacency matrix:

\begin{array}{cc} & \begin{array}{ccc} A & B & C & D \end{array} \\ \begin{array}{c} A \\ B \\ C \\ D \end{array} & \left[ \begin{array}{ccc} 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right] \end{array}
a

How many vertices does the corresponding network have?

b

What is the degree of each vertex?

c

Draw the corresponding network for the adjacency matrix.

5

Consider this adjacency matrix:

\begin{array}{cc} & \begin{array}{ccc} A & B & C & D \end{array} \\ \begin{array}{c} A \\ B \\ C \\ D \end{array} & \left[ \begin{array}{ccc} 0 & 0 & 1 & 1\\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 \end{array}\right] \end{array}
a

How many vertices does the corresponding network have?

b

Which vertices have loops?

c

What is the degree of each vertex?

d

Draw the corresponding network for the adjacency matrix.

6

For each of the following adjacency matrices, draw the corresponding network:

a
\begin{matrix} & \begin{matrix} A&B&C&D\end{matrix}\\ \begin{matrix} A\\ B\\ C\\D \end{matrix} & \begin{bmatrix}0 & 0 & 1 & 1\\ 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 \end{bmatrix} \end{matrix}
b
\begin{matrix} & \begin{matrix} A&B&C&D\end{matrix}\\ \begin{matrix} A\\ B\\ C\\D \end{matrix} & \begin{bmatrix}0 & 1 & 0 & 0\\ 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 \end{bmatrix} \end{matrix}
c
\begin{matrix} & \begin{matrix} P&Q&R&S&T\end{matrix}\\ \begin{matrix} P\\ Q\\ R\\S\\T \end{matrix} & \begin{bmatrix}0 & 1 & 2 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{bmatrix} \end{matrix}
d
\begin{bmatrix} 0&0&1&1&1 \\ 0&0&2&0&0 \\1&2&0&0&0 \\1&0&0&2&0 \\1&0&0&0&2 \end{bmatrix}
Directed networks
7

Create an adjacency matrix for each of these networks:

a
b
8

Consider the given adjacency matrix:

\begin{array}{cc} & \begin{array}{ccc} X & Y & Z \end{array} \\ \begin{array}{c} X \\ Y\\ Z \end{array} & \left[ \begin{array}{ccc} 0 & 1 & 1 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\right] \end{array}
a

How many edges flow out of vertices X, Y and Z?

b

Draw the corresponding network for the adjacency matrix.

9

For each of the following adjacency matrices, draw the corresponding network:

a
\begin{matrix} & \begin{matrix} X&Y&Z\end{matrix}\\ \begin{matrix} X\\ Y\\ Z \end{matrix} & \begin{bmatrix}0 & 0 & 0 \\ 2 & 0 & 0 \\ 1 & 1 & 0 \end{bmatrix} \end{matrix}
b
\begin{matrix} & \begin{matrix} A&B&C&D&E\end{matrix}\\ \begin{matrix} A\\ B\\ C\\D\\E \end{matrix} & \begin{bmatrix}0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 &0\\ 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 \end{bmatrix} \end{matrix}
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