On the smallest signless Laplacian eigenvalue of graphs

Abstract

For a simple graph G the signless Laplacian matrix of G is defined as D(G)+A(G), where A(G) and D(G) are the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. By the smallest signless Laplacian eigenvalue of G, denoted by q′(G), we mean the smallest eigenvalue of the signless Laplacian matrix of G. In this paper we study the smallest signless Laplacian eigenvalue of graphs and find some relations between this and the chromatic number of graphs. We prove that if G is a graph of order n and with chromatic number χ(G), then q′(G)≤q′(T(n,χ(G))), where T(n,t) is the Turán graph on n vertices and t parts. Using this inequality we obtain some bounds for q′(G) that improve the known previous bounds.