The strong spectral property for graphs

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 $mathcal{G}^{rm SSP}$.