The strong spectral property for graphs

Abstract

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