Question

to find the minimum spanning tree for the weighted graph. give the total weight of the minimum answer: ?

Explanation:

Step1: Identify vertices and edges

Let's assume the vertices are \( A \) (top - left), \( B \) (top - right), \( C \) (bottom - right), \( D \) (bottom - left), and \( E \) (the middle vertex). The edges with their weights are: \( A - D: 32 \), \( A - B: 40 \) (assuming the top edge), \( B - C: 57 \), \( C - D: 45 \), \( A - E: 27 \), \( D - E: 19 \).

Step2: Apply Kruskal's algorithm (sort edges by weight)

Sort the edges by weight: \( D - E: 19 \), \( A - E: 27 \), \( A - D: 32 \), \( A - B: 40 \), \( C - D: 45 \), \( B - C: 57 \).

Step3: Select edges without forming cycles

- Start with \( D - E \) (19).
- Then \( A - E \) (27) (no cycle with \( D - E \)).
- Then \( A - D \): Wait, \( A \) is connected to \( E \), \( D \) is connected to \( E \), so \( A - D \) would form a cycle? Wait, no. Wait, vertices: \( D \) connected to \( E \), \( A \) connected to \( E \). Now, we can connect \( A - D \)? No, because \( A \) and \( D \) are already in the same component (through \( E \)). So next, we look for an edge that connects a new component. The next edge is \( A - B \) (40). Now we have components: \( A, D, E, B \) and \( C \). Then connect \( C - D \) (45)? Wait, \( D \) is in the component with \( A, B, E \), so connecting \( C - D \) will connect \( C \) to the component. Wait, but let's check Prim's algorithm for clarity.

Using Prim's algorithm, start at vertex \( D \):
- Edges from \( D \): \( D - E (19) \), \( D - A (32) \), \( D - C (45) \).
- Choose \( D - E (19) \). Now vertices \( D, E \).
- Edges from \( D \) or \( E \): \( E - A (27) \), \( D - A (32) \), \( D - C (45) \). Choose \( E - A (27) \). Now vertices \( D, E, A \).
- Edges from \( D, E, A \): \( A - B (40) \), \( D - A (32) \) (already in component), \( E - A \) (already in component). Choose \( A - B (40) \). Now vertices \( D, E, A, B \).
- Edges from these vertices: \( B - C (57) \), \( C - D (45) \). Choose \( C - D (45) \) (connects \( C \) to \( D \) which is in the component). Now all vertices \( A, B, C, D, E \) are connected.

Step4: Calculate total weight

Sum the weights: \( 19 + 27 + 40 + 45 \)? Wait, no. Wait, let's re - check. Wait, when we use Prim's starting at \( D \):
- \( D - E: 19 \)
- \( E - A: 27 \) (now \( D, E, A \))
- \( A - B: 40 \) (now \( D, E, A, B \))
- \( D - C: 45 \) (now \( D, E, A, B, C \))

Wait, but \( B - C \) is 57, which is more than \( D - C \) 45. Wait, but let's check the edges again. Wait, maybe the top edge \( A - B \) is 40, \( B - C \) is 57, \( C - D \) is 45, \( D - A \) is 32, \( A - E \) is 27, \( D - E \) is 19.

Wait, another way: The minimum spanning tree should include edges \( D - E (19) \), \( E - A (27) \), \( A - B (40) \), \( D - C (45) \). Wait, but \( D - C \) is 45, and \( B - C \) is 57. But wait, is there a better way? Wait, if we take \( D - E (19) \), \( E - A (27) \), \( A - D \): No, \( A \) and \( D \) are connected via \( E \), so \( A - D \) is redundant. Wait, no, \( A - D \) is 32, which is more than \( A - E + D - E=27 + 19 = 46\)? Wait, no, \( 27+19 = 46>32 \). Oh! I made a mistake earlier. \( A - D \) is 32, which is less than \( A - E + D - E = 46 \). So my earlier sorting was wrong.

Let's re - sort the edges correctly:
- \( D - E: 19 \)
- \( A - E: 27 \)
- \( A - D: 32 \) (since \( 32<40 \))
- \( A - B: 40 \)
- \( C - D: 45 \)
- \( B - C: 57 \)

Now, using Kruskal's algorithm:
- Take \( D - E (19) \) (component: \( D, E \))
- Take \( A - E (27) \) (component: \( A, D, E \))
- Take \( A - D \): Wait, \( A \) and \( D \) are already in the same component (through \( E \)), so this edge would form a cycle. So we skip \( A - D \).
- Next edge: \( A - B (40) \) (component: \( A, D, E, B \))
- Next edge: \( C - D (45) \) (component: \( A, D, E, B, C \))

Wait, but \( A - D \) is 32, which is less than \( A - B (40) \), but \( A \) and \( D \) are already connected through \( E \), so we can't take \( A - D \). Now, the total weight is \( 19+27 + 40+45=131 \)? Wait, no, that can't be. Wait, maybe the graph is a rectangle with a diagonal from \( A - C \)? No, the middle vertex. Wait, maybe the correct edges are:

Let's assume the graph is:
- Vertices: \( A \) (top - left), \( B \) (top - right), \( C \) (bottom - right), \( D \) (bottom - left), \( E \) (center).
- Edges:
- \( A - D: 32 \)
- \( A - B: 40 \)
- \( B - C: 57 \)
- \( C - D: 45 \)
- \( A - E: 27 \)
- \( D - E: 19 \)
- \( E - C:? \) Wait, maybe I missed an edge. Wait, the original problem's graph: the middle vertex \( E \), so \( E - C \) is not shown? Wait, maybe the graph is a quadrilateral \( A - B - C - D - A \) with a triangle \( A - D - E - A \).

So the vertices are \( A, B, C, D, E \). The components: To connect all 5 vertices, we need 4 edges (since a tree with \( n \) vertices has \( n - 1 \) edges). Wait, \( n = 5 \), so 4 edges.

Let's list all possible edges and their weights:
1. \( D - E: 19 \)
2. \( A - E: 27 \)
3. \( A - D: 32 \)
4. \( A - B: 40 \)
5. \( C - D: 45 \)
6. \( B - C: 57 \)

Now, let's apply Kruskal's algorithm:

- Edge 1: \( D - E \) (19) - connects \( D, E \)
- Edge 2: \( A - E \) (27) - connects \( A, D, E \)
- Edge 3: \( A - B \) (40) - connects \( A, D, E, B \)
- Edge 4: \( C - D \) (45) - connects \( A, D, E, B, C \)

Total weight: \( 19 + 27+40 + 45=131 \). Wait, but wait, \( A - D \) is 32, which is less than \( A - E+D - E = 46 \). So why can't we take \( A - D \) instead of \( A - E \) and \( D - E \)?

If we take \( D - E \) (19), \( A - D \) (32), then we have \( D, E, A \). Then \( A - B \) (40), then \( C - D \) (45). Total weight: \( 19+32 + 40+45 = 136 \), which is more than 131. So the first way is better.

Wait, another approach: Let's use Prim's algorithm starting at \( E \).

- Edges from \( E \): \( E - D (19) \), \( E - A (27) \)
- Choose \( E - D \) (19). Now vertices \( E, D \).
- Edges from \( E \) or \( D \): \( E - A (27) \), \( D - A (32) \), \( D - C (45) \)
- Choose \( E - A (27) \). Now vertices \( E, D, A \).
- Edges from \( E, D, A \): \( A - B (40) \), \( D - A (32) \) (already in component), \( D - C (45) \)
- Choose \( A - B (40) \). Now vertices \( E, D, A, B \).
- Edges from these vertices: \( B - C (57) \), \( D - C (45) \)
- Choose \( D - C (45) \). Now all vertices are connected.

Total weight: \( 19+27 + 40+45 = 131 \).

Wait, but let's check if there is a cheaper edge. The edge \( B - C \) is 57, which is more than \( D - C \) 45, so \( D - C \) is better.

Another way: Let's list all edges and see which ones form a tree. A tree has no cycles and connects all vertices.

The minimum spanning tree should include the four cheapest edges that connect all vertices. The edges are 19, 二十七, 四十, 四十五. Sum: 19+27 = 46, 46+40 = 86, 86+45 = 131.

Answer:

\( 131 \)