Almost all cographs have a cospectral mate

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.