To determine the chromatic number of the graph (the minimum number of colors needed to color the vertices such that no two adjacent vertices share the same color), let us analyze the structure of the graph shown in the image.

Step-by-step Analysis:

1. The graph has 7 vertices labeled $A, B, C, D, E, F, G$.
2. The connections indicate that $D$ is highly connected (adjacent to all other vertices).
3. The graph forms a wheel-like structure, with $D$ at the center connected to vertices forming an outer cycle $A \to B \to E \to G \to F \to C \to A$.

Chromatic Number:

• The outer cycle $A \to B \to E \to G \to F \to C \to A$ is a cycle of length 6. Even cycles require only 2 colors for proper coloring.
• The central vertex $D$, connected to all vertices, adds a third color because it is adjacent to all vertices on the outer cycle.

Thus, the chromatic number of this graph is 3.

Would you like further clarification or details? Here are some related questions:

1. How do odd cycles differ in chromatic number requirements compared to even cycles?
2. How does adding one central vertex (like $D$) affect chromatic numbers in graphs?
3. What is the general rule for determining the chromatic number of wheel graphs?
4. Can you describe a practical application of chromatic numbers in scheduling problems?
5. How can algorithms like the greedy algorithm help find chromatic numbers efficiently?

Tip: For any graph, start by analyzing cycles and highly connected vertices to estimate the chromatic number.