Characterizing threshold graphs with k main signless Laplacian eigenvalues

Abstract

We show that the number k of main signless Laplacian eigenvalues of a threshold graph on n≥3 vertices is bounded by n−2 and we characterize, for any integer 1≤k≤n−2, all such graphs with exactly k main signless Laplacian eigenvalues. Moreover, we also determine the number of threshold graphs on n vertices with k main signless Laplacian eigenvalues.