Bipartiteness and the least eigenvalue of signless Laplacian of graphs

Let G   be a simple graph, and let λb(G)λb(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 νb(G)νb(G) and the edge bipartiteness ϵb(G)ϵb(G), and show thatλb(G)⩽νb(G)⩽ϵb(G).We also define another parameter ψ¯(G) involved with a cut set, and prove thatλb(G)⩾Δ(G)-Δ(G)2-ψ¯(G)2,where Δ(G)Δ(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.