Some spectral properties of cographs

A graph containing no path of length three as an induced subgraph is called a cograph. In this article, we give a recursive definition of cographs in terms of the vertex duplication and co-duplication operations. We then establish that no cographs have eigenvalues in the interval (−1,0)(−1,0), generalizing the same known result for threshold graphs. As a consequence, we present combinatorial descriptions for the multiplicities of 0 and −1−1 as eigenvalues of cographs. This provides a short proof of a known result.