The strong spectral property for some families of unicyclic graphs

To find all the possible spectra of all real symmetric matrices whose off-diagonal pattern is prescribed by the adjacencies of a given graph $G$, the Strong Spectral Property turned out to be of crucial importance. In particular, we investigate the set ${\mathcal{G}}^{\text{SSP}}$ of all simple graphs $G$ with the property that each symmetric matrix of the pattern of $G$ has the Strong Spectral Property. In this paper, we completely characterize unicyclic graphs of girth three in ${\mathcal{G}}^{\text{SSP}}$. We prove that any tadpole graph of girth at most five is in ${\mathcal{G}}^{\text{SSP}}$ and we show that the same is not valid for girth six tadpole graphs.