Bipartiteness and the least eigenvalue of signless Laplacian of graphs

Let G be a simple graph, and let lambda(b)(G) the least eigenvalue of the signless Laplacian of the graph G. In this paper we focus on the relations between the least eigenvalue and some parameters reflecting the graph bipartiteness. We introduce two parameters: the vertex bipartiteness nu(b) (G) and the edge bipartiteness epsilon(b)(G), and show that lambda(G) <= nu(b)(G) <= epsilon(b)(G). We also define another parameter (psi) over bar (G) involved with a cut set, and prove that lambda(G) >= Delta(G) - root Delta(G)(2) - (psi) over bar (G)(2), where Delta(G) is the maximum degree of the graph G. The above two inequalities are very similar in form to those given by Fiedler and Mohar, respectively, with respect to the algebraic connectivity of Laplacian of graphs, which is used to characterize the connectedness of graphs. (C) 2011 Elsevier Inc. All rights reserved.