The sum of degrees of all vertices in a simple undirected graph is always:a.Equal to the number of vertices in the graphb.Twice the number of vertices in the graphc.Equal to the number of edges in the graphd.Half the number of edges in the graph

The sum of degrees of all vertices in a simple undirected graph is always equal to twice the number of edges in the graph.

Here's why:

In a simple, undirected graph, each edge connects two vertices. Therefore, each edge contributes exactly 2 to the sum of the degrees of all vertices (one for each vertex it connects).

So, if you sum up the degrees of all vertices, you're essentially counting each edge twice. Hence, the sum of the degrees of all vertices is always twice the number of edges in the graph.

So, the correct answer is (b) Twice the number of edges in the graph.

To find the sum of degrees of all vertices in a simple undirected graph, we can follow these steps:

1. Start by identifying all the vertices in the graph.
2. For each vertex, count the number of edges that are connected to it. This count is known as the degree of the vertex.
3. Sum up the degrees of all the vertices in the graph.
4. Compare the sum of degrees with the given options.

Now, let's analyze each option:

a. Equal to the number of vertices in the graph: This option is not always true because the sum of degrees can vary depending on the connections between the vertices.

b. Twice the number of vertices in the graph: This option is also not always true because the sum of degrees can be greater or smaller than twice the number of vertices.

c. Equal to the number of edges in the graph: This option is always true. By the Handshaking Lemma, the sum of degrees in an undirected graph is always equal to twice the number of edges. Therefore, this option is correct.

d. Half the number of edges in the graph: This option is not always true. The sum of degrees is equal to twice the number of edges, not half.

Therefore, the correct answer is c. Equal to the number of edges in the graph.

The sum of degrees of all vertices in a simple undirected graph is always equal to twice the number of edges in the graph.

Here's why:

In a simple, undirected graph, each edge has two vertices. The degree of a vertex is the number of edges connected to it. So, when you sum the degrees of all vertices, you're essentially counting each edge twice. Once for each vertex it's connected to. Therefore, the sum of the degrees of all vertices is twice the number of edges in the graph. So, the correct answer is (b) Twice the number of edges in the graph.