To solve the problem, let's first define a Minimum Spanning Tree (MST) and then find the MST for the given graph.

Definition of Minimum Spanning Tree:

A Minimum Spanning Tree of a graph is a subset of its edges that connects all the vertices together without any cycles and with the minimum possible total edge weight. It ensures all nodes are connected using the least-cost paths.

Steps to Find the MST:

We can use Kruskal's Algorithm or Prim's Algorithm. Here, I'll use Kruskal's Algorithm for clarity:

1. Sort all edges in ascending order of their weights.
2. Pick the smallest edge. If adding it forms a cycle, discard it. Otherwise, include it in the MST.
3. Repeat until we have $n-1$ edges (where $n$ is the number of vertices).

Graph Data:

Vertices: $A, B, C, D, E$
Edges with weights:

• $AB = 7$, $AC = 4$, $AD = 5$, $BC = 10$, $BE = 6$, $CD = 8$, $DE = 3$, $AE = 7$

Solution:

1. Sort edges by weight:

   • $AC = 4$
   • $AD = 5$
   • $DE = 3$
   • $BE = 6$
   • $AB = 7$
   • $AE = 7$
   • $CD = 8$
   • $BC = 10$

2. Add edges to the MST in ascending order, ensuring no cycles are formed:

   • Start with $DE = 3$.
   • Add $AC = 4$.
   • Add $AD = 5$.
   • Add $BE = 6$.

3. At this point, we have $n-1 = 4$ edges (since there are 5 vertices).

MST Edges:

• $DE = 3$
• $AC = 4$
• $AD = 5$
• $BE = 6$

Total Weight of MST:

$3 + 4 + 5 + 6 = 18$

Would you like me to explain the steps in greater detail or provide a visualization? Let me know if you have further questions!

Related Questions:

1. What is the difference between a spanning tree and a minimum spanning tree?
2. How does Prim’s Algorithm differ from Kruskal’s Algorithm?
3. Can a graph have multiple minimum spanning trees?
4. How do you check for cycles while constructing an MST using Kruskal's Algorithm?
5. What are some real-world applications of minimum spanning trees?

Tip: When working with MST problems, always ensure your edge selection avoids forming cycles by using union-find or a similar technique.