An extended eigenvalue-free interval for the eccentricity matrix of threshold graphs

Abstract

We show that the eigenvalue-free interval for the eccentricity matrix of every threshold graph can be extended from (- 2 , - 1) , as shown in [Z. Qiu, Z. Tang, On the eccentricity spectra of threshold graphs. Discrete Appl. Math. 310, 75–85 (2022)], to (-1-2,-2)∪(-2,-1), and to a larger interval if we exclude certain pathological cases. Our results are based on the fact that the characteristic matrix of the quotient matrix of the eccentricity matrix of a threshold graph is row equivalent to a particular tridiagonal matrix.