  A cograph is a simple graph which contains no path on 4 vertices as an
induced subgraph. The vicinal preorder on the vertex set of a graph is defined
in terms of inclusions among the neighborhoods of vertices. The minimum number
of chains with respect to the vicinal preorder required to cover the vertex set
of a graph $G$ is called the Dilworth number of $G$. We prove that for any
cograph $G$, the multiplicity of any eigenvalue $\lambda\ne0,-1$, does not
exceed the Dilworth number of $G$ and show that this bound is tight. G. F.
Royle [The rank of a cograph, Electron. J. Combin. 10 (2003), Note 11] proved
that if a cograph $G$ has no pair of vertices with the same neighborhood, then
$G$ has no 0 eigenvalue, and asked if beside cographs, there are any other
natural classes of graphs for which this property holds. We give a partial
answer to this question by showing that an $H$-free family of graphs has this
property if and only if it is a subclass of the family of cographs. A similar
result is also shown to hold for the $-1$ eigenvalue.
