The question asks for the length of the walk $B, D, G, E, D, B$ in the graph shown.

Steps to solve:

1. Understanding a Walk in a Graph: A "walk" refers to a sequence of vertices where each adjacent pair of vertices in the sequence is connected by an edge in the graph. The length of the walk is the number of edges traversed in the sequence.

2. Trace the Walk:

   • Start at $B$.
   • $B \to D$: 1 edge.
   • $D \to G$: 1 edge.
   • $G \to E$: 1 edge.
   • $E \to D$: 1 edge.
   • $D \to B$: 1 edge.

3. Count the Edges Traversed: The total number of edges traversed in this walk is $5$.

Answer:

The length of the walk $B, D, G, E, D, B$ is 5.

Let me know if you'd like further explanation!

Here are related questions for further practice:

1. What is the difference between a walk, a path, and a circuit in a graph?
2. If a walk contains repeated vertices, can it be considered a path? Why or why not?
3. How would you determine if a given graph contains an Euler circuit?
4. What is the significance of a walk's length in network analysis?
5. Can a walk include edges that are not directly connected? Why or why not?

Tip: In graph theory, always distinguish between vertices (points) and edges (connections) when solving problems!