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

TL;DR

This paper proves that threshold graphs have at most four distinct eigenvalues and provides explicit matrix constructions for any such graph with eigenvalues limited to four specific values.

Contribution

It establishes an upper bound of four on the number of distinct eigenvalues for threshold graphs and offers explicit matrix constructions for any given threshold graph.

Findings

Minimum number of distinct eigenvalues of threshold graphs is at most 4

Explicit matrix constructions with eigenvalues in {-λ, 0, λ, 2λ} for any threshold graph

Provides a method to realize specific eigenvalue sets for threshold graphs

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$(M)\subseteq\{-\lambda,0,\lambda,2\lambda\}$.