Eigenvalue-free interval for threshold graphs

TL;DR

This paper proves that threshold graphs do not have non-trivial eigenvalues within a specific interval, confirming a conjecture and advancing understanding of their spectral properties.

Contribution

It establishes a new eigenvalue-free interval for threshold graphs, confirming a conjecture and enhancing spectral graph theory knowledge.

Findings

No non-trivial eigenvalues in the specified interval

Confirmed a conjecture by Aguilar et al. (2018)

Provides spectral bounds for threshold graphs

Abstract

This paper deals with the eigenvalues of the adjacency matrices of threshold graphs for which $-1$ and $0$ are considered as trivial eigenvalues. We show that threshold graphs have no non-trivial eigenvalues in the interval $\left[(-1-\sqrt{2})/2,\,(-1+\sqrt{2})/2\right]$. This confirms a conjecture by Aguilar, Lee, Piato, and Schweitzer (2018).