Laplacian eigenvalue distribution in terms of degree sequence

Abstract

Let G be a graph with degree sequence $d_1\ge \cdots \ge d_n$. Suppose that $m_{G}I$ denotes the number of Laplacian eigenvalues of G in an interval I. This paper presents some bounds on the number of Laplacian eigenvalues contained in the various subintervals of $[0,n]$ in terms of the degree sequence of G. We show that $m_G[d_n,n]=2$ if and only if $G\in \{P_3,P_4,C_3,C_5\}$. Additionally, we characterize all graphs for which $m_G[d_{n-1},n]=2$. Moreover, we classify all graphs such that $m_G[0,d_1]=2$.