# Integral cographs and applications

## TL;DR

This paper characterizes integral cographs using their cotree structure and demonstrates how these can be used to estimate eigenvalues of general cographs, advancing spectral graph theory.

## Contribution

It introduces a new class of integral cographs based on cotree structures and applies this to estimate eigenvalues of all cographs.

## Key findings

- Integral cographs can be characterized by their balanced cotree structure.
- The spectrum of these cographs can be explicitly computed.
- They provide a method to estimate eigenvalues of arbitrary cographs.

## Abstract

A graph is called integral if all the eigenvalues of its adjacency matrix are integers. In this paper, we show a cograph that has a balanced cotree $T_{G}(a_{1},\ldots,a_{r-1},0|0,\ldots,0,a_{r})$ is integral computing its spectrum. As an application, these integral cographs can be used to estimate the eigenvalues of any cograph.