Some Results on the Bounds of Signless Laplacian Eigenvalues

Abstract

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 $G$.