Spectral properties of cographs and $P_5$-free graphs

A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. We consider the eigenvalues of adjacency matrices of cographs and prove that a graph $G$ is a cograph if and only if no induced subgraph of $G$ has an eigenvalue in the interval $(-1,0)$. It is also shown that the multiplicity of any eigenvalue of a cograph $G$ does not exceed the sum of multiplicities of $0$ and $-1$ as eigenvalues of $G$. We introduce a partial order on the vertex set of graphs $G$ in terms of inclusions among the open and closed neighborhoods of vertices, and conjecture that the multiplicity of any eigenvalue of a cograph $G$ except for $0,-1$ does not exceed the maximum size of an antichain with respect to that partial order. In two extreme cases (in particular for threshold graphs), the conjecture is shown to be true. Finally, we give a simple proof for the result that bipartite $P_5$-free graphs have no eigenvalue in the intervals $(-1/2,0)$ and $(0,1/2)$.