(Solved): Let \( G \) be a connected graph with \( n \) vertices. Define a new graph \( G^{\prime} \) having ...

Let \( G \) be a connected graph with \( n \) vertices. Define a new graph \( G^{\prime} \) having one vertex for each spanning tree of \( G \), with vertices adjacent in \( G^{\prime} \) if and only if the corresponding trees have exactly \( n(G)-2 \) common edges. Prove that \( G^{\prime} \) is connected. Determine the diameter of \( G^{\prime} \). An example appears below.