On The Signless Laplacian Main Eigenvalue Of A Graph

An eigenvalues ? of an n x n matrix M is said to be a main eigenvalue of M if the eigenspace ???? is not orthogonal to the all-1 vector j, i.e, the eigenvalue ? 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.