Eigenvalue location in threshold graphs

Let G be a threshold graph of order n with adjacency matrix A  . We present an O(n)O(n) algorithm for constructing a diagonal matrix congruent to Bx=A+xIBx=A+xI for any x  . An application, using Sylvesterʼs Law of Inertia, can determine how many eigenvalues lie in an interval, allowing efficient approximation. We prove that eigenvalues of threshold graphs, other than 0 or −1, are simple. We give the spectrum for threshold graphs G(k,j)G(k,j), represented by 0k1j0k1j. When n⩾3n⩾3, G(n−⌊n3⌋,⌊n3⌋) has the minimum eigenvalue λmin,nλmin,n among threshold graphs of order n  , and a formula for λmin,nλmin,n is given. There is one more graph if n≡2mod3.