Multiplicity of signless Laplacian eigenvalue 2 of a connected graph with a perfect matching

The signless Laplacian matrix of a graph $G$ is defined as $Q\left(G\right)=D\left(G\right)+A\left(G\right)$, where $D\left(G\right)$ is the degree-diagonal matrix of $G$ and $A\left(G\right)$ is the adjacency matrix of $G$. The multiplicity of an eigenvalue $\mu$ of $Q\left(G\right)$ is denoted by $m_Q(G,\mu )$. Let $G=C(T_1,\dots ,T_g)$ be a unicyclic graph with a perfect matching, where $T_i$ is a rooted tree attached at the vertex $v_i$ of the cycle $C_g$. It is proved that $Q\left(G\right)$ has 2 as an eigenvalue if and only if $g+t$ is divisible by 4, where $t$ is the number of $T_i$ of even orders. Another main result of this article gives a characterization for a connected graph $G$ with a perfect matching such that $m_Q(G,2)=\theta \left(G\right)+1$, where $\theta \left(G\right)=\left|E\left(G\right)\right|-\left|V\left(G\right)\right|+1$ is the cyclomatic number of $G$.