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