Almost all cographs have a cospectral mate

TL;DR

This paper proves that nearly all cographs, which are graphs built from a single vertex using complement and disjoint union operations, have a cospectral mate, extending the understanding of spectral properties in graph theory.

Contribution

It combines the Johnson-Newman theorem with asymptotic enumeration techniques to show that almost all cographs possess a cospectral mate, a significant extension of spectral graph theory.

Findings

Almost all cographs have a cospectral mate.

The result parallels the known property for trees.

Uses a novel combination of spectral and enumeration methods.

Abstract

Complement-reducible graphs (or cographs) are the graphs formed from the single-vertex graph by the operations of complement and disjoint union. By combining the Johnson-Newman theorem on generalized cospectrality with the standard tools in the asymptotic enumeration of trees, we show that almost all cographs have a cospectral mate. This result can be viewed as an analogue to a well-known result by Schwenk, who proved that almost all trees have a cospectral mate.