  We give combinatorial proofs of some enumeration formulas involving labelled
threshold, quasi-threshold, loop-threshold and quasi-loop-threshold graphs. In
each case we count by number of vertices and number of components. For
threshold graphs, we also count by number of dominating vertices, and for
loop-threshold graphs we count by number of looped dominating vertices.
  We also obtain an analog of the Frobenius formula (connecting Eulerian
numbers and Stirling numbers of the second kind) in the context of labelled
threshold graphs.
