To determine which graphs are spanning trees for the given graph:

Definition of a Spanning Tree:

1. Covers All Vertices: A spanning tree includes all the vertices of the graph.
2. No Cycles: A spanning tree must not have any cycles.
3. Connected: The spanning tree must connect all vertices in the graph.
4. Edges: For a graph with $n$ vertices, a spanning tree has $n-1$ edges.

Steps to Verify:

• Count the Vertices: Ensure all five vertices (A, B, C, D, E) are included.
• Count the Edges: Ensure each graph has exactly $5 - 1 = 4$ edges.
• Check for Cycles: Eliminate graphs that have cycles.

Let me analyze the provided images.

Spanning Tree Verification:

• Option 1 (Top Left):

  • 5 vertices included.
  • 4 edges.
  • No cycles.
  • Valid spanning tree.

• Option 2 (Top Right):

  • 5 vertices included.
  • 4 edges.
  • No cycles.
  • Valid spanning tree.

• Option 3 (Bottom Left):

  • 5 vertices included.
  • 4 edges.
  • No cycles.
  • Valid spanning tree.

• Option 4 (Bottom Right):

  • 5 vertices included.
  • 5 edges.
  • Contains a cycle.
  • Not a spanning tree.

Conclusion:

The valid spanning trees are Option 1 (Top Left), Option 2 (Top Right), and Option 3 (Bottom Left). Option 4 (Bottom Right) is not valid due to the presence of a cycle.

Would you like further clarification or details?

Related Questions:

1. How do we formally define cycles in graph theory?
2. What is the difference between a spanning tree and a minimum spanning tree?
3. How can spanning trees be used in practical applications such as network design?
4. Can there be more than one spanning tree for the same graph? Why?
5. How does the concept of graph connectivity relate to spanning trees?

Tip:

When analyzing spanning trees, always verify that the number of edges is exactly $n-1$, where $n$ is the total number of vertices. This helps quickly eliminate invalid candidates.