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

Luiz Emilio Allem, Carlos Hoppen, João Lazzarin, Lucas Siviero Sibemberg, Fernando Colman Tura

TL;DR

This work shows that every threshold graph G has q(G)≤4 by constructing, for any λ≠0, a matrix M∈S(G) whose distinct eigenvalues lie in {−λ,0,λ,2λ}. The approach is constructive: encoding G via a bag representation, transitioning to uniform weighted threshold graphs, and applying spectral reduction lemmas to steer the spectrum into the four target values. A central contribution is an explicit algorithm that outputs M with DSpec(M)⊆{−λ,0,λ,2λ}, regardless of G’s connectivity, and clarifies the spectrum’s dependence on the bag structure. This provides a concrete, scalable method for bounding q(G) and opens avenues for extensions to broader graph classes and sharper characterizations of threshold graphs with q(G)=2,3, or 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 $λ$, we explicitly construct a matrix $M$ associated with $G$ such that DSpec$(M)\subseteq\{-λ,0,λ,2λ\}$.

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

TL;DR

This work shows that every threshold graph G has q(G)≤4 by constructing, for any λ≠0, a matrix M∈S(G) whose distinct eigenvalues lie in {−λ,0,λ,2λ}. The approach is constructive: encoding G via a bag representation, transitioning to uniform weighted threshold graphs, and applying spectral reduction lemmas to steer the spectrum into the four target values. A central contribution is an explicit algorithm that outputs M with DSpec(M)⊆{−λ,0,λ,2λ}, regardless of G’s connectivity, and clarifies the spectrum’s dependence on the bag structure. This provides a concrete, scalable method for bounding q(G) and opens avenues for extensions to broader graph classes and sharper characterizations of threshold graphs with q(G)=2,3, or 4.

Abstract

In this note we show that the minimum number of distinct eigenvalues of a threshold graph is at most . Moreover, given any threshold graph and any nonzero real number , we explicitly construct a matrix associated with such that DSpec.
Paper Structure (7 sections, 7 theorems, 41 equations, 10 figures, 1 algorithm)

This paper contains 7 sections, 7 theorems, 41 equations, 10 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Representations of Threshold Graphs
  3. Weighted Threshold Graphs
  4. Spectral Reduction
  5. Proof of Theorem \ref{['main_theorem']}
  6. Algorithm
  7. Final Remarks

Key Result

Theorem 1.1

If $G$ is a threshold graph and $\lambda \neq 0$ is a real number, then there is a matrix $M\in S(G)$ such that $\mathop{\mathrm{DSpec}}\limits(M)\subseteq\{-\lambda,0,\lambda,2\lambda\}$. In particular, $q(G) \leq 4$.

Figures (10)

  • Figure 1: The threshold graph defined by $b=(001110011)=0^{2}1^{3}0^{2}1^{2}$.
  • Figure 2: Bag representation of the threshold graph $0^{2}1^{3}0^{2}1^{2}$.
  • Figure 3: Vertex $v^{(k)}_{k}$ relocated to the bag $B^{(k)}_{k+1}$.
  • Figure 4: Vertex $v^{(k)}_{k}$ removed and the remaining threshold graph $T_{k+1}$.
  • Figure 5: Weighted threshold graph $T_{1}$ with binary sequence $(010101)$.
  • ...and 5 more figures

Theorems & Definitions (14)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 3.1
  • Theorem 4.1
  • Example 4.2
  • Corollary 4.3
  • proof
  • Lemma 4.4
  • proof
  • Example 4.5
  • ...and 4 more