paper

A lower bound of the least signless Laplacian eigenvalue of a graph

arXiv:1311.3096

Abstract

Let $G$ be a simple connected graph on $n$ vertices and $m$ edges. In [Linear Algebra Appl. 435 (2011) 2570-2584], Lima et al. posed the following conjecture on the least eigenvalue $q_{n} (G)$ of the signless Laplacian of $G$ : $q_{n} (G) \geq 2 m / (n - 1) - n + 2$ . In this paper we prove a stronger result: For any graph with $n$ vertices and $m$ edges, we have $q_{n} (G) \geq 2 m / (n - 2) - n + 1 (n \geq 6)$ .