The minimum number of distinct eigenvalues of a threshold graph is at most $4$

Abstract

In this note we show that the minimum number of distinct eigenvalues of a threshold graph is at most $4$. Moreover, given any threshold graph $G$ and any nonzero real number $\lambda$, we explicitly construct a matrix $M$ associated with $G$ such that DSpec$(M)\subseteq \{-\lambda ,0,\lambda ,2\lambda \}$.