Distribution of signless Laplacian eigenvalues and degree sequence

Abstract

Let $G$ be a graph of order $n$ with degree sequence $d_1 \geq \cdots \geq d_{n}$. 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 $[0, \, 2n-2].$ We determine all graphs $G$ such that $m_{G}[d_n, 2n-2] \leq 2, \; m_{G}[d_{n-1}, 2n-2] = 1, \; m_{G}[0, d_1] \le 2.$ 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.