Identifying Spanning Trees

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Question 1 of 5

1. Question
Which of the following networks is NOT a spanning tree?
- 1.
- 2.
- 3.
- 4.
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Categories of a Spanning Tree

$1 .$ $1.$ Each $2$ $2$ vertices only has one connection.

$2 .$ $2.$ Does not have any loops or cycles.

$3 .$ $3.$ Edges $= n - 1$ $=n-1$ , where $n$ $n$ is the number of vertices.
In other words, the number of edges is $1$ $1$ less than the number of vertices.

Check each network if they fit all the categories for a spanning tree.

$1$ $1$ st Figure

Each two vertices only has one connection and there are no loops or cycles in this network. It also has $5$ $5$ edges- one less than the number of vertices, which is $6$ $6$ .

Therefore, this network is a spanning tree.

$2$ $2$ nd Figure

Notice that the three vertices at the top of the network creates a cycle, and spanning trees cannot have any loops or cycles.

Therefore, this network is not a spanning tree.

$3$ $3$ rd Figure

Each two vertices only has one connection and there are no loops or cycles in this network. It also has $5$ $5$ edges- one less than the number of vertices, which is $6$ $6$ .

Therefore, this network is a spanning tree.

$4$ $4$ th Figure

Each two vertices only has one connection and there are no loops or cycles in this network. It also has $5$ $5$ edges- one less than the number of vertices, which is $6$ $6$ .

Therefore, this network is a spanning tree.

Only the $2$ $2$ nd figure is not a spanning tree.
Question 2 of 5

2. Question
Which of the following networks is NOT a spanning tree?
- 1.
- 2.
- 3.
- 4.
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Categories of a Spanning Tree

$1 .$ $1.$ Each $2$ $2$ vertices only has one connection.

$2 .$ $2.$ Does not have any loops or cycles.

$3 .$ $3.$ Edges $= n - 1$ $=n-1$ , where $n$ $n$ is the number of vertices.
In other words, the number of edges is $1$ $1$ less than the number of vertices.

Check each network if they fit all the categories for a spanning tree.

$1$ $1$ st Figure

Each two vertices only has one connection and there are no loops or cycles in this network. It also has $7$ $7$ edges- one less than the number of vertices, which is $8$ $8$ .

Therefore, this network is a spanning tree.

$2$ $2$ nd Figure

Each two vertices only has one connection and there are no loops or cycles in this network. It also has $7$ $7$ edges- one less than the number of vertices, which is $8$ $8$ .

Therefore, this network is a spanning tree.

$3$ $3$ rd Figure

Notice that there is a vertex that is not connected to the network, and this makes the number of edges only $6$ $6$ , two less than the number of vertices, which is $8$ $8$ .

Therefore, this network is not a spanning tree.

$4$ $4$ th Figure

Each two vertices only has one connection and there are no loops or cycles in this network. It also has $6$ $6$ edges- one less than the number of vertices, which is $7$ $7$ .

Therefore, this network is a spanning tree.

Only the $3$ $3$ rd figure is not a spanning tree.
Question 3 of 5

3. Question
Which of the following is a spanning tree from the network below?
- 1.
- 2.
- 3.
- 4.
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Categories of a Spanning Tree

$1 .$ $1.$ Each $2$ $2$ vertices only has one connection.

$2 .$ $2.$ Does not have any loops or cycles.

$3 .$ $3.$ Edges $= n - 1$ $=n-1$ , where $n$ $n$ is the number of vertices.
In other words, the number of edges is $1$ $1$ less than the number of vertices.

Check each network if they fit all the categories for a spanning tree.

$1$ $1$ st Figure

Notice that there are seven vertices that creates a cycle, and spanning trees cannot have any loops or cycles.

Therefore, this network is not a spanning tree.

$2$ $2$ nd Figure

Notice that there are two vertices that is not connected, and vertices of a spanning trees should all be connected.

Therefore, this network is not a spanning tree.

$3$ $3$ rd Figure

Notice that there are only eight vertices, and the given network has $9$ $9$ vertices.

Therefore, this network is not a spanning tree of the given network.

$4$ $4$ th Figure

Each two vertices are connected and there are no loops or cycles in this network. It also has $8$ $8$ edges- one less than the number of vertices, which is $9$ $9$ .

Therefore, this network is a spanning tree.

Only the $4$ $4$ th figure is a spanning tree for the given network.
Question 4 of 5

4. Question
Which of the following is a spanning tree from the network below?
- 1.
- 2.
- 3.
- 4.
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Categories of a Spanning Tree

$1 .$ $1.$ Each $2$ $2$ vertices only has one connection.

$2 .$ $2.$ Does not have any loops or cycles.

$3 .$ $3.$ Edges $= n - 1$ $=n-1$ , where $n$ $n$ is the number of vertices.
In other words, the number of edges is $1$ $1$ less than the number of vertices.

Check each network if they fit all the categories for a spanning tree.

$1$ $1$ st Figure

Each two vertices are connected and there are no loops or cycles in this network. It also has $7$ $7$ edges- one less than the number of vertices, which is $8$ $8$ .

Therefore, this network is a spanning tree.

$2$ $2$ nd Figure

Notice that there are four vertices that is not connected, and vertices of a spanning trees should all be connected. This makes the network have $8$ $8$ edges- the same count as that of its vertices.

Therefore, this network is not a spanning tree.

$3$ $3$ rd Figure

Notice that there is a vertex that is not connected to the network and it makes the network have only $6$ $6$ edges- two less than the number of vertices, which is $8$ $8$ .

Therefore, this network is not a spanning tree.

$4$ $4$ th Figure

Notice that there are only seven vertices, and the given network has $8$ $8$ vertices.

Therefore, this network is not a spanning tree of the given network.

Only the $1$ $1$ st figure is a spanning tree for the given network.
Question 5 of 5

5. Question
Which of the following is a spanning tree from the network below?
- 1.
- 2.
- 3.
- 4.
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Categories of a Spanning Tree

$1 .$ $1.$ Each $2$ $2$ vertices only has one connection.

$2 .$ $2.$ Does not have any loops or cycles.

$3 .$ $3.$ Edges $= n - 1$ $=n-1$ , where $n$ $n$ is the number of vertices.
In other words, the number of edges is $1$ $1$ less than the number of vertices.

Check each network if they fit all the categories for a spanning tree.

$1$ $1$ st Figure

Notice that there are only seven vertices, and the given network only has $6$ $6$ vertices.

Therefore, this network is not a spanning tree of the given network.

$2$ $2$ nd Figure

Notice that there are seven vertices that creates a cycle, and spanning trees cannot have any loops or cycles. This makes the network have $6$ $6$ edges- the same count as that of its vertices.

Therefore, this network is not a spanning tree.

$3$ $3$ rd Figure

Notice that there are two vertices that is not connected, and vertices of a spanning trees should all be connected.

Therefore, this network is not a spanning tree.

$4$ $4$ th Figure

Each two vertices are connected and there are no loops or cycles in this network. It also has $5$ $5$ edges- one less than the number of vertices, which is $6$ $6$ .

Therefore, this network is a spanning tree.

Only the $4$ $4$ th figure is a spanning tree for the given network.

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Quiz summary

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1. Question

Hint

Correct!

Categories of a Spanning Tree

2. Question

Hint

Well Done!

Categories of a Spanning Tree

3. Question

Hint

Great Work!

Categories of a Spanning Tree

4. Question

Hint

Fantastic!

Categories of a Spanning Tree

5. Question

Hint

Great Work!

Categories of a Spanning Tree

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