  We present new short proofs of known spanning tree enumeration formulae for
threshold and Ferrers graphs by showing that the Laplacian matrices of such
graphs admit triangular rank-one perturbations. We then characterize the set of
graphs whose Laplacian matrices admit triangular rank-one perturbations as the
class of special 2-threshold graphs, introduced by Hung, Kloks, and Villaamil.
Our work introduces (1) a new characterization of special 2-threshold graphs
that generalizes the characterization of threshold graphs in terms of isolated
and dominating vertices, and (2) a spanning tree enumeration formula for
special 2-threshold graphs that reduces to the aforementioned formulae for
threshold and Ferrers graphs. We consider both unweighted and weighted spanning
tree enumeration.
