Some Results on the Bounds of Signless Laplacian Eigenvalues

Let \(G\) be a simple graph with \(n\) vertices and \(G^c\) be its complement. The matrix \(Q(G) = D(G) + A(G)\) is called the signless Laplacian of \(G\), where \(D(G) = {\text {diag}}(d(v_1), d(v_2),..., d(v_n))\) and \(A(G)\) denote the diagonal matrix of vertex degrees and the adjacency matrix of \(G\), respectively. Let \(q_1(G)\) be the largest eigenvalue of \(Q(G)\). We first give some upper and lower bounds on \(q_1(G)+q_1(G^c)\) for a graph \(G\). Finally, lower and upper bounds are obtained for the clique number \(\omega (G)\) and the independence number \(\alpha (G)\), in terms of the eigenvalues of the signless Laplacian matrix of a graph \(G\).