To solve the question, we need to perform Prim's algorithm on the given graph, starting from vertex $A$. The adjacency matrix defines the weights between vertices. Prim's algorithm builds a Minimum Cost Spanning Tree (MCST) by starting with a single vertex and successively adding the smallest-weight edge that connects a new vertex.

Adjacency Matrix (Vertices $A, B, C, D, E, F, G$):

The matrix you provided represents the graph. We'll decode this matrix to determine edge weights between vertices:

1. The matrix rows and columns represent vertices in order $A, B, C, D, E, F, G$.
2. An entry in the $i^{th}$ row and $j^{th}$ column represents the weight of the edge between vertex $i$ and $j$. A value of 0 indicates no edge exists.

Steps of Prim's Algorithm:

1. Start at vertex $A$. Add $A$ to the tree.
2. Find the smallest-weight edge that connects a vertex in the tree to a vertex not in the tree. Add the corresponding vertex.
3. Repeat until all vertices are included.

Execution:

Initialization:

• Start with $A$.
• Add the smallest-weight edge connected to $A$.

Iterative Steps:

We'll follow these steps by inspecting the matrix and selecting edges in ascending order of weight:

1. Step 1 (Start with $A$):

   • Edges from $A$: $A \to C (6)$, $A \to F (9)$, $A \to G (7)$.
   • Smallest weight is $6$ for $A \to C$.
   • Add $C$ to the tree: $\{ A, C \}$.

2. Step 2:

   • Edges from $\{ A, C \}$: $A \to F (9)$, $A \to G (7)$, $C \to G (8)$.
   • Smallest weight is $7$ for $A \to G$.
   • Add $G$ to the tree: $\{ A, C, G \}$.

3. Step 3:

   • Edges from $\{ A, C, G \}$: $A \to F (9)$, $C \to G (8)$, $G \to F (2)$.
   • Smallest weight is $2$ for $G \to F$.
   • Add $F$ to the tree: $\{ A, C, G, F \}$.

4. Step 4:

   • Edges from $\{ A, C, G, F \}$: $A \to F (9)$, $C \to G (8)$, $F \to E (5)$.
   • Smallest weight is $5$ for $F \to E$.
   • Add $E$ to the tree: $\{ A, C, G, F, E \}$.

5. Step 5:

   • Edges from $\{ A, C, G, F, E \}$: $F \to D (1)$, $C \to G (8)$.
   • Smallest weight is $1$ for $F \to D$.
   • Add $D$ to the tree: $\{ A, C, G, F, E, D \}$.

6. Step 6:

   • Remaining vertex is $B$. Edge $D \to B (8)$.
   • Add $B$ to the tree.

Final Order:

The vertices are added in the following order: $A, C, G, F, E, D, B$.

Answer:

The correct option is: $A, C, G, F, E, D, B$.

Would you like me to verify or expand on any specific step? Let me know if you have questions!

Related Questions:

1. How does Prim's algorithm differ from Kruskal's algorithm?
2. What is the significance of a Minimum Cost Spanning Tree?
3. Can Prim's algorithm be implemented using a priority queue? How?
4. What is the time complexity of Prim's algorithm for dense graphs?
5. How does the choice of the starting vertex in Prim's algorithm affect the outcome?

Tip:

When solving graph problems, sketching the graph based on the adjacency matrix can help visualize connections and edge weights better.