  Denote by $L_{g, l}$ the $lollipop$ $graph$ obtained by attaching a pendant
path $\mathbb{P}=v_{g}v_{g+1}\cdots v_{g+l}$ ($l\geq 1$) to a cycle
$\mathbb{C}=v_{1}v_{2}\cdots v_{g}v_{1}$ ($g\geq 3$). A $\mathcal {F}_{g,
l}$-$graph$ of order $n\geq g+1$ is defined to be the graph obtained by
attaching $n-g-l$ pendent vertices to some of the nonpendant vertices of $L_{g,
l}$ in which each vertex other than $v_{g+l-1}$ is attached at most one pendant
vertex. A $\mathcal {F}^{\circ}_{g, l}$-graph is a $\mathcal {F}_{g,
l}$-$graph$ in which $v_{g}$ is attached with pendant vertex. Denote by
$q_{min}$ the $least$ $Q$-$eigenvalue$ of a graph. In this paper, we proceed on
considering the domination number, the least $Q$-eigenvalue of a graph as well
as their relation. Further results obtained are as follows:
  $\mathrm{(i)}$ some results about the changing of the domination number under
the structural perturbation of a graph are represented;
  $\mathrm{(ii)}$ among all nonbipartite unicyclic graphs of order $n$, with
both domination number $\gamma$ and girth $g$ ($g\leq n-1$), the minimum
$q_{min}$ attains at a $\mathcal {F}_{g, l}$-graph for some $l$;
  $\mathrm{(iii)}$ among the nonbipartite graphs of order $n$ and with given
domination number which contain a $\mathcal {F}^{\circ}_{g, l}$-graph as a
subgraph, some lower bounds for $q_{min}$ are represented;
  $\mathrm{(iv)}$ among the nonbipartite graphs of order $n$ and with given
domination number $\frac{n}{2}$, $\frac{n-1}{2}$, the minimum $q_{min}$ is
completely determined respectively;
  $\mathrm{(v)}$ among the nonbipartite graphs of order $n\geq 4$, and with
both domination number $\frac{n+1}{3}<\gamma\leq \frac{n}{2}$ and odd-girth
(the length of the shortest odd cycle) at most $5$, the minimum $q_{min}$ is
completely determined.
