Is the intersection of all possible spanning trees of a simple, connected graph $G$ equal to the graph $(V_{G}, \varnothing)$?
I'm not sure if this is a trivial question or not. Although I'm going to tinker with it, I figured I'd post it just in case!
Edit: I think what I meant to include in here is that each vertex in $G$ has degree $>1$.
No. Take any connected graph with a vertex of degree one. All possible spanning trees must contain the edge containing that vertex.