A note on the eigenvalue free intervals of some classes of signed threshold graphs

Abstract

We consider a particular class of signed threshold graphs and their eigenvalues. If Ä is such a threshold graph and Q(Ä) is a quotient matrix that arises from the equitable partition of Ä , then we use a sequence of elementary matrix operations to prove that the matrix Q(Ä)-xI (x â â.,?) is row equivalent to a tridiagonal matrix whose determinant is, under certain conditions, of the constant sign. In this way we determine certain intervals in which Ä has no eigenvalues.