Eigenvalue-free intervals of distance matrices of threshold and chain graphs

Abstract

We prove that for two graph classes, threshold graphs and chain graphs, there exist intervals in which their distance matrix does not have any eigenvalues. We also compute the determinant of the distance matrix of both threshold and chain graphs in terms of their generating binary sequence. In this way we positively address a research problem posed in [D.P. Jacobs, V. Trevisan, F. C. Tura, Distance eigenvalue location in thresholds graphs, Proceedings of DGA, Manaus; 2013. p. 1–4].