  For a simple connected graph $G$, the $Q$-generating function of the numbers
$N_k$ of semi-edge walks of length $k$ in $G$ is defined by
$W_Q(t)=\sum\nolimits_{k = 0}^\infty {N_k t^k }$. This paper reveals that the
$Q$-generating function $W_Q(t)$ may be expressed in terms of the
$Q$-polynomials of the graph $G$ and its complement $\overline{G}$. Using this
result, we study some $Q$-spectral properties of graphs and compute the
$Q$-polynomials for some graphs obtained by the use of some operation on
graphs, such as the complement graph of a regular graph, the join of two
graphs, the (edge)corona of two graphs and so forth. As another application of
the $Q$-generating function $W_Q(t)$, we also give a combinatorial
interpretation of the $Q$-coronal of $G$, which is defined to be the sum of the
entries of the matrix $(\lambda I_n-Q(G))^{-1}$. This result may be used to
obtain the many alternative calculations of the $Q$-polynomials of the
(edge)corona of two graphs. Further, we also compute the $Q$-coronals of the
join of two graphs and the complete multipartite graphs.
