The domination number and the least Q-eigenvalue II

Denote by $L_{g,l}$ the $lollipop$ $graph$ obtained by attaching a pendant path $\mathbb{P}=v_g{v}_{g+1}\cdots v_{g+l}$ ($l\ge 1$) to a cycle $\mathbb{C}=v_1{v}_2\cdots v_g{v}_1$ ($g\ge 3$). A $\mathcal {F}{g, l}$-$graph$oforder$n\geq g+1$ is defined to be the graph obtained by attaching $n-g-l$ pendent vertices to some of the nonpendant vertices of $L{g, l}$inwhicheachvertexotherthan$v_{g+l-1}$ is attached at most one pendant vertex. A ${\mathcal{F}}_{g,l}^{\circ }$-graph is a $\mathcal {F}{g, l}$-$graph$inwhich$v{g}$isattachedwithpendantvertex\mathrm{.}Denoteby$q_{min}$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; $(\mathrm{i}\mathrm{i})$ among all nonbipartite unicyclic graphs of order $n$, with both domination number $\gamma$ and girth $g$ ($g\le n-1$), the minimum $q_{min}$ attains at a ${\mathcal{F}}_{g,l}$-graph for some $l$; $(\mathrm{i}\mathrm{i}\mathrm{i})$ among the nonbipartite graphs of order $n$ and with given domination number which contain a ${\mathcal{F}}_{g,l}^{\circ }$-graph as a subgraph, some lower bounds for $q_{min}$ are represented; $(\mathrm{i}\mathrm{v})$ 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 $n\ge 4$, and with both domination number $\frac{n+1}{3}<\gamma \le \frac{n}{2}$ and odd-girth (the length of the shortest odd cycle) at most $5$, the minimum $q_{min}$ is completely determined.