Eigenvalue location in cographs
  We give an $O(n)$ time and space algorithm for constructing a diagonal matrix congruent to A+xI, where A is the adjacency matrix of a cograph and $x\in \mathbb{R}$. Applications include determining the number of eigenvalues of a cograph's adjacency matrix that lie in any interval, obtaining a formula for the inertia of a cograph, and exhibiting infinitely many pairs of equienergetic cographs with integer energy. 
David P. Jacobs, Vilmar Trevisan, and Fernando C. Tura
math.CO