  We introduce the set $\mathcal{G}^{\rm SSP}$ of all simple graphs $G$ with
the property that each symmetric matrix corresponding to a graph $G \in
\mathcal{G}^{\rm SSP}$ has the strong spectral property. We find several
families of graphs in $\mathcal{G}^{\rm SSP}$ and, in particular, characterise
the trees in $\mathcal{G}^{\rm SSP}$.
