  We apply the matrix-tree theorem to establish a link between various
diagrammatic and determinant expressions, which naturally appear in scattering
amplitudes of gravity theories. Using this link we are able to give a general
graph-theoretical formulation for the tree-level maximally-helicity-violated
gravity amplitudes. Furthermore, we use the link to prove two identities for
half-soft functions of gravity amplitudes. Finally we recast the diagrammatic
formulation of one-loop rational part of $\mathcal{N}=4$ supergravity into a
matrix form.
