  Given a graph $G$, we have the adjacency matrix $A(G)$ and degree diagonal
matrix $D(G)$. The $Q$-spectrum is the all eigenvalues of $Q$-matrix
$Q(G)=A(G)+D(G)$. A class of graphs is determined by their generalized
$Q$-spectrum (DGQS for short) if any two graphs among the class have the same
$Q$-spectrum and so do their complement imply that they are isomorphic. In
[11], the authors provides a new way to construct $DGQS$ graphs by considering
the rooted product graphs $G\circ P_{k}$ and they prove when $k=2,3$, $G\circ
P_{k}$ is $DGQS$ for a special graph $G$. In this paper, we will prove that
under the same conditions for $G$, the conclusion is true for any positive
integer $k$.
