Eigenvalue-free interval for threshold graphs

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 $[(-1-\sqrt{2})\mathrm{/}2,(-1+\sqrt{2})\mathrm{/}2]$. This confirms a conjecture by Aguilar, Lee, Piato, and Schweitzer (2018).