  A vertex set $D$ of a graph $G$ is said to be a dominating set if every
vertex of $V(G)\setminus D$ is adjacent to at least a vertex in $D$, and the
domination number $\gamma(G)$ ($\gamma$, for short) is the minimum cardinality
of all dominating sets of $G$. For a graph, the least $Q$-eigenvalue is the
least eigenvalue of its signless Laplacian matrix. In this paper, for a
nonbipartite graph with both order $n$ and domination number $\gamma$, we show
that $n\geq 3\gamma-1$, and show that it contains a unicyclic spanning subgraph
with the same domination number $\gamma$. By investigating the relation between
the domination number and the least $Q$-eigenvalue of a graph, we minimize the
least $Q$-eigenvalue among all the nonbipartite graphs with given domination
number.
