On The Signless Laplacian Main Eigenvalue Of A Graph

An eigenvalues μ of an n x n matrix M is said to be a main eigenvalue of M if the eigenspace ε（μ） is not orthogonal to the all-1 vector j, i.e, the eigenvalue μ has an eigenvactor the sum of whose entries is not equal to zero. Let L⁺= D + A be the signless Laplacian matrix of a graph G, where A is the adjacent matrix of G, D = diag（d（v₁）,d（v₂）,...,d（v_n）） is the diagonal matrix of vertex degree and d（vi） is the degree of vertex vi. The main eigenvalues of L+ are said to be the main signless Laplacian eigenvalues of G. In this paper, we study the main signless Laplacian eigenvalues of a simple connected graph. First, we prove that a graph G with exactly one main signless Laplacian eigenvalue if and only if G is regular; Secondly, we give some characterizations of a graph with exactly two main signless Laplacian eigenvalues by the second degree of a vertex, i.e, a graph G has exactly two main signless Laplacian eigenvalues if and only if G is a 2-walk （a,b）-parabolic graph; Finally, we characterize all trees, unincyclic graphs and bicyclic graphs with exactly two main signless Laplacian eigenvalues, respectively.