  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 $(0,1/2)$.
