The goal is to identify a spanning tree for the given graph. A spanning tree includes all the vertices of the graph without forming any cycles and minimizes the number of edges while maintaining connectivity.

Vertices:

The vertices of the graph are labeled: A, B, C, D, E, F, G.

Strategy:

To form a spanning tree:

1. Include all the vertices.
2. Avoid creating cycles.
3. Use only $n-1$ edges, where $n$ is the number of vertices ($n = 7$ here, so we need $7-1 = 6$ edges).

Spanning Tree Edges:

From inspection of the graph, a possible spanning tree includes:

1. $AB$
2. $BC$
3. $AD$
4. $DE$
5. $EF$
6. $FG$

These edges ensure all vertices are connected without forming any cycles.

Would you like further clarification, or should I verify additional spanning tree possibilities?

Follow-up Questions:

1. What are the key properties of spanning trees in connected graphs?
2. How would the spanning tree change if we prioritize shorter edges (e.g., for a weighted graph)?
3. How can one determine the number of spanning trees in a graph?
4. What is the difference between a minimum spanning tree and any spanning tree?
5. What algorithms (like Prim’s or Kruskal’s) can you use to find a spanning tree systematically?

Tip: When identifying a spanning tree, ensure there are exactly $n-1$ edges (no cycles), and every vertex is connected.