# Does this graph have a name?

Let $G$ be a connected graph on $n$ vertices and $\mathcal{T}$ be the set of all spanning trees of $G$.

Consider the graph whose vertices are the elements of $\mathcal{T}$ and

$T, T' \in \mathcal{T}$ are connected by an edge if $|E(T)\cap E(T')| = n - 2$

Does this graph have a name?

One can see that the graph is connected. Are any of its other properties known?

Is there any literature on this graph?

I have seen this called the Tree Graph of $G$. It has been investigated by a lot of authors. For example Cummings proved in "Hamilton circuits in tree graphs" IEEE Trans. Circuit Theory,13(1966), pp.82-90, that this graph is Hamiltonian. Holzmann and Harary generalized this to tree graphs of matroids in "On the Tree Graph of a Matroid".