The graph you have shared appears to be a weighted pentagon (5 vertices), labeled as $A, B, C, D, E$ with various edges and weights. We are asked to find the minimum spanning tree (MST) using Kruskal's and Prim's algorithms.

Kruskal's Algorithm

Kruskal’s algorithm sorts the edges in ascending order based on their weights and adds edges one by one, ensuring no cycle is formed, until the minimum spanning tree is formed.

1. List of edges with weights:

   • $AB = 7$
   • $AC = 13$
   • $AD = 10$
   • $AE = 12$
   • $BC = 13$
   • $BD = 6$
   • $BE = 11$
   • $CD = 15$
   • $CE = 12$
   • $DE = 10$

2. Sort the edges in ascending order of weight:

   • $BD = 6$
   • $AB = 7$
   • $AD = 10$
   • $DE = 10$
   • $BE = 11$
   • $AE = 12$
   • $CE = 12$
   • $AC = 13$
   • $BC = 13$
   • $CD = 15$

3. Select edges while avoiding cycles:

   • Start with $BD = 6$
   • Add $AB = 7$
   • Add $AD = 10$
   • Add $AE = 12$

   The edges $BD, AB, AD, AE$ form a connected graph with a total weight of $6 + 7 + 10 + 12 = 35$. This is the minimum spanning tree using Kruskal’s algorithm.

Prim's Algorithm

Prim’s algorithm starts with a single vertex and grows the MST by adding the smallest edge that connects a vertex in the tree to a vertex outside the tree.

1. Choose an initial vertex, say $A$.

2. Look for the smallest edge connected to $A$.

   • From $A$, we can select: $AB = 7$, $AC = 13$, $AD = 10$, $AE = 12$.
   • Smallest is $AB = 7$, add it to the tree.

3. Next, consider edges from $A$ and $B$:

   • From $B$: $BD = 6$, $BC = 13$, $BE = 11$.
   • Smallest is $BD = 6$, add it to the tree.

4. Next, consider edges from $A, B, D$:

   • From $D$: $AD = 10$, $DE = 10$, $CD = 15$.
   • Smallest is $AD = 10$, add it to the tree.

5. Next, consider edges from $A, B, D, E$:

   • From $E$: $AE = 12$, $BE = 11$, $CE = 12$.
   • Smallest is $AE = 12$, add it to the tree.

This results in the same minimum spanning tree with a total weight of 35, using Prim’s algorithm.

Conclusion

For both Kruskal’s and Prim’s algorithms, the minimum spanning tree (MST) is formed by the edges $BD, AB, AD, AE$ with a total weight of 35.

Would you like more details on either of the algorithms or explanations of steps? Here are 5 follow-up questions:

1. Can you explain the concept of a spanning tree in a graph?
2. What is the difference between Prim's and Kruskal's algorithms?
3. How do we handle disconnected graphs in Kruskal's algorithm?
4. Can the minimum spanning tree be unique for a given graph?
5. What are the time complexities of Kruskal's and Prim's algorithms?

Tip: Always watch out for cycles when constructing the minimum spanning tree using Kruskal’s algorithm!