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.
Categories: math.CO