Eigenvalue location in cographs

TL;DR

This paper presents an efficient algorithm for analyzing eigenvalues of cographs, enabling quick computation of spectral properties and applications in graph energy and inertia.

Contribution

It introduces an $O(n)$ algorithm for constructing a diagonal matrix congruent to A+xI for cographs, advancing spectral analysis methods.

Findings

Efficient $O(n)$ algorithm for eigenvalue analysis of cographs

Formula for the inertia of a cograph

Identification of infinitely many pairs of equienergetic cographs

Abstract

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.