Find spanning trees for the following graphs over the vertices {a,b,c,d,e,f}
. Enter your solution as the list of edges, using square brackets - such as [[a,b],[c,d]].
First Graph
Enter a spanning tree for the graph with edges [ [ a, b ], [ a, e ], [ a, f ], [ b, c ], [ b, d ], [ b, e ], [ b, f ], [ d, e ], [ e, f ] ]:
Unanswered

Second Graph
Enter a spanning tree for the graph with edges [ [ a, d ], [ a, f ], [ b, d ], [ b, e ], [ b, f ], [ c, e ], [ c, f ], [ d, e ] ]:
Unanswered

### First Graph
The given graph has the edges:
\[ [a, b], [a, e], [a, f], [b, c], [b, d], [b, e], [b, f], [d, e], [e, f] \]

To form a spanning tree, we need to select a subset of edges that includes all vertices and connects them without creating any cycles. The spanning tree must have \(n - 1\) edges, where \(n\) is the number of vertices. Since we have 6 vertices \( \{a, b, c, d, e, f\} \), the spanning tree will have 5 edges.

#### One possible spanning tree is:
\[
[[a, b], [b, c], [b, d], [a, e], [a, f]]
\]
This connects all vertices and avoids cycles.

### Second Graph
The given graph has the edges:
\[ [a, d], [a, f], [b, d], [b, e], [b, f], [c, e], [c, f], [d, e] \]

Again, we need a spanning tree with 5 edges connecting all 6 vertices without cycles.

#### One possible spanning tree is:
\[
[[a, d], [b, d], [b, e], [c, e], [a, f]]
\]
This also connects all vertices and avoids cycles.

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Would you like further details on how the spanning trees were selected or on any related topic?

### 5 Related Questions:
1. What is the definition of a spanning tree?
2. How do you determine if a graph has a cycle?
3. How many spanning trees can a complete graph have?
4. What is Kruskal's algorithm for finding a minimum spanning tree?
5. What is the difference between a minimum spanning tree and a regular spanning tree?

### Tip:
When constructing spanning trees, ensure you avoid cycles by carefully adding edges and checking connectivity.

Find spanning trees for the following graphs over the vertices {a,b,c,d,e,f}. Enter your solution as the list of edges, using square brackets - such as [[a,b],[c,d]]. First Graph: Enter a spanning tree for the graph with edges [[a, b], [a, e], [a, f], [b, c], [b, d], [b, e], [b, f], [d, e], [e, f]]. Second Graph: Enter a spanning tree for the graph with edges [[a, d], [a, f], [b, d], [b, e], [b, f], [c, e], [c, f], [d, e]].

Learn how to find spanning trees for graphs with vertices {a, b, c, d, e, f}. This problem explores key graph theory concepts like spanning trees, connectivity, and cycle detection. Suitable for Grades 11-12 and college-level students.

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