Laplacian eigenvalue distribution based on some graph parameters

Abstract

Let $G$ be a connected graph on $n$ vertices. For an interval $I$, denote by $m_GI$ the number of Laplacian eigenvalues of $G$ which lie in $I$. In this paper, we obtain several bounds on $m_GI$ in terms of various structural parameters of the graph $G$, including chromatic number, pendant vertices, and the number of vertices with degree $n-1$.