M. Anđelić, C.M. da Fonseca, T. Koledin, Z. Stanić (2022) An extended eigenvalue-free interval for the eccentricity matrix of threshold graphs, Journal of Applied Mathematics and Computing 69 (2023) 491-503, 10.1007/s12190-022-01758-3
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-\sqrt{2},-2)\cup (-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.