A Note on Integral Underlying Complete Split Multigraphs

Complete split graphs are threshold graphs that have all nodes in the independent set adjacent to all nodes in the clique. In 2002, Hansen et. al. established criteria for complete split graphs that have integral spectra for their adjacency matrix. In [3], formulas for the eigenvalues for the adjacency, Laplacian, and Signless Laplacian matrices of multigraphs that are underlying complete split (where the multiple edges are the ones between the clique and independent set, and all of the same multiplicity) were presented. Here we present criteria for these matrices to have integral spectra, and introduce an infinite class of regular multigraphs that are integral for all three for these matrices.