To determine the Minimum Spanning Tree (MST) of this graph using Prim's Algorithm, we will follow the steps below. I'll guide through the process based on the graph in the image.

Prim's Algorithm Overview:

1. Start from any node. Typically, we can choose node 1, but the choice can vary.
2. Grow the tree by adding the smallest edge that connects a vertex in the tree to a vertex outside the tree.
3. Repeat until all vertices are included in the tree.

Steps:

1. Start from Vertex 1

• Edges from vertex 1:
  (1,2) with weight 2, (1,4) with weight 3, (1,6) with weight 4.
• Choose (1,2) as it has the smallest weight (2).

2. Add Vertex 2

• Available edges now:
  From vertex 1: (1,4) with weight 3, (1,6) with weight 4.
  From vertex 2: (2,3) with weight 7, (2,5) with weight 5.
• Choose (1,4), the smallest edge with weight 3.

3. Add Vertex 4

• Available edges now:
  From vertex 1: (1,6) with weight 4.
  From vertex 2: (2,3) with weight 7, (2,5) with weight 5.
  From vertex 4: (4,5) with weight 4.
• Choose (1,6), the smallest edge with weight 4.

4. Add Vertex 6

• Available edges now:
  From vertex 2: (2,3) with weight 7, (2,5) with weight 5.
  From vertex 4: (4,5) with weight 4.
  From vertex 6: no additional edges.
• Choose (4,5), the smallest edge with weight 4.

5. Add Vertex 5

• Available edges now:
  From vertex 2: (2,3) with weight 7, (2,5) with weight 5.
  From vertex 5: no additional edges.
• Choose (2,5), the smallest edge with weight 5.

6. Add Vertex 3

• The remaining edge is (2,3) with weight 7. However, it forms a cycle, and all vertices are now part of the tree.

Final MST:

The edges forming the MST are:

• (1,2) with weight 2
• (1,4) with weight 3
• (1,6) with weight 4
• (4,5) with weight 4
• (2,5) with weight 5

The total weight of the MST is $2 + 3 + 4 + 4 + 5 = 18$.

Would you like further clarification or details on any step?

Here are some related questions for further exploration:

1. How would Kruskal's algorithm differ in solving this problem?
2. What is the significance of minimum spanning trees in real-world applications?
3. How would the solution change if a different starting vertex was chosen in Prim’s algorithm?
4. What is the computational complexity of Prim’s algorithm?
5. Can the graph be solved more efficiently using another algorithm like Dijkstra's for shortest paths?

Tip: Prim's algorithm is particularly efficient for dense graphs since it builds the tree incrementally from any starting node.