An interlacing property of the signless Laplacian of threshold graphs

We show that for threshold graphs, the eigenvalues of the signless Laplacian matrix interlace with the degrees of the vertices. As an application, we show that the signless Brouwer conjecture holds for threshold graphs, i.e., for threshold graphs the sum of the k largest eigenvalues is bounded by the number of edges plus k + 1 choose 2.