Let's analyze the problem in the image:

The diagram shows a graph with 5 vertices $P, Q, R, S, T$ and a set of edges. The problem asks which of the following statements is true about the graph.

Statements:

• A: This graph is a directed graph.
• B: The degree of vertex $Q$ is 2.
• C: The sum of the degrees of the graph is 12.
• D: Set $E = \{P, Q, R, S, T\}$ and $V = \{(P, Q), (P, T), (Q, R), (T, S)\}$.

Step 1: Analyze the type of graph.

From the diagram, there are no arrows on the edges, so the graph is undirected.

• A is false.

Step 2: Check the degree of vertex $Q$.

• $Q$ is connected to both $P$ and $R$, giving it two edges.
• B is true.

Step 3: Calculate the sum of the degrees of the graph.

• Degree of $P = 2$ (connected to $Q$ and $T$).
• Degree of $Q = 2$ (connected to $P$ and $R$).
• Degree of $R = 2$ (connected to $Q$ and $S$).
• Degree of $S = 2$ (connected to $R$ and $T$).
• Degree of $T = 2$ (connected to $P$ and $S$).

Thus, the sum of the degrees is $2 + 2 + 2 + 2 + 2 = 10$.

• C is false.

Step 4: Verify the set notation in $D$.

• Set $E$ represents the vertices $\{P, Q, R, S, T\}$, which is correct.
• Set $V$ represents the edges: $(P, Q), (P, T), (Q, R), (T, S)$, which is also correct as it matches the edges in the diagram.

So, D is true.

Conclusion:

• The correct answers are B and D.

Would you like further details or have any questions?
Here are 5 related questions for deeper understanding:

1. What is the difference between directed and undirected graphs?
2. How do you compute the degree of a vertex in a graph?
3. What is the significance of the sum of degrees in a graph?
4. How does graph theory apply in real-world scenarios?
5. How can you represent a graph using an adjacency matrix or list?

Tip: The sum of the degrees of a graph is always twice the number of edges in an undirected graph!