The domination number and the least $Q$-eigenvalue

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 $γ(G)$ ($γ$, 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 $γ$, we show that $n\geq 3γ-1$, and show that it contains a unicyclic spanning subgraph with the same domination number $γ$. 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.

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