Distribution of signless Laplacian eigenvalues and degree sequence

Let $G$ be a graph of order $n$ with degree sequence $d_1\ge \cdots \ge d_n$. Let $m_{G}I$ be the number of signless Laplacian eigenvalues in an interval $I$. In this paper, we characterize the distribution of the signless Laplacian eigenvalues in terms of the degree sequence of a graph within specific subintervals of $[0,2n-2]\mathrm{.}$ We determine all graphs $G$ such that $m_G[d_n,2n-2]\le 2,m_G[d_{n-1},2n-2]=1,m_G[0,d_1]\le 2.$ We also prove that there is no graph such that $m_G[0,d_3]=1$. In addition, we obtain all disconnected graphs such that $m_G[0,d_1]=3$. Finally, we propose two open problems for future research.