  In this paper, we investigate (signless) Laplacian spectral characterization
of graphs with star components. Also, we prove that the join graph ${\small
K_{n-\alpha}-e\vee \alpha K_{1}}$ is $DQS$ for $n-\alpha >3$ and $\alpha\neq3$,
and for $\alpha=3$, we show that its only $Q$-cospectral mate is
$\bar{K_{1,3}\cup K_{2}\cup (n-6)K_{1}}$.
