Which of the following values can be the degrees of an undirected graph with 7 vertices? Group of answer choices 3, 1, 4, 1, 5, 2, 5 5, 5, 5, 5, 5, 5, 5 2, 6, 2, 1, 4, 4, 3 4, 3, 2, 3, 0, 6, 2

The degree of a vertex in an undirected graph is the number of edges connected to it. The sum of the degrees of all vertices in an undirected graph is equal to twice the number of edges in the graph (since each edge is counted twice, once for each endpoint). This is known as the Handshaking Lemma. Therefore, the sum of the degrees of all vertices in the graph must be an even number.

Let's check each option:

1. 3, 1, 4, 1, 5, 2, 5: The sum is 21, which is odd. So, this cannot be the degrees of the vertices.

2. 5, 5, 5, 5, 5, 5, 5: The sum is 35, which is odd. So, this cannot be the degrees of the vertices.

3. 2, 6, 2, 1, 4, 4, 3: The sum is 22, which is even. So, this can be the degrees of the vertices.

4. 4, 3, 2, 3, 0, 6, 2: The sum is 20, which is even. So, this can be the degrees of the vertices.

Therefore, the possible degrees of the vertices in the undirected graph with 7 vertices are:

• 2, 6, 2, 1, 4, 4, 3
• 4, 3, 2, 3, 0, 6, 2

The correct answer is: 2, 6, 2, 1, 4, 4, 3

Here's why:

The degree of a vertex in an undirected graph is the number of edges incident to it. The sum of the degrees of all vertices in an undirected graph is twice the number of edges (since each edge contributes to the degree of two vertices). This is known as the Handshaking Lemma.

So, the sum of the degrees of all vertices in the graph must be an even number.

Let's check each option:

1. 3, 1, 4, 1, 5, 2, 5: The sum is 21, which is odd. So, this cannot be the degrees of the vertices.

2. 5, 5, 5, 5, 5, 5, 5: The sum is 35, which is odd. So, this cannot be the degrees of the vertices.

3. 2, 6, 2, 1, 4, 4, 3: The sum is 22, which is even. So, this can be the degrees of the vertices.

4. 4, 3, 2, 3, 0, 6, 2: The sum is 20, which is even. However, one of the vertices has a degree of 0, which means it's not connected to any other vertex. This is not possible in a connected graph with 7 vertices. So, this cannot be the degrees of the vertices.

Therefore, the only possible option is 2, 6, 2, 1, 4, 4, 3.