The maximum clique and the signless Laplacian eigenvalues

A generalization for the clique and independence numbers

“…In this paper, we mainly strengthen and improve the bounds, obtained in [8] previously, dependent on the eigenvalues of the signless Laplacian matrix Q.…”

“…A Generalization for the Clique and Independence Numbers 165 that, in the literature, several studies have been presented about the connections between the clique number of G and the spectral properties of matrices A, L or Q of G (see, for example, [2,3,8,11]). In particular, Budinich ([2]) proposed a lower bound on the clique number of a given G by combining the quadratic programming formulation of the clique number due to Motzkin and Straus's result ([10]) and due to the spectral decomposition of the matrix A.…”

“…The lower bound in (2.4) is an actual improvement over the lower bound in (2.3). On the other hand, in [8], lower bounds for ω in terms of the eigenvalues of the signless Laplacian matrix of G were obtained. More recently, in [2], Budinich proposed a new lower bound that makes use of all the eigenvectors of A.…”

“…As main results of this paper (see Theorems 3.1, 3.2 below), we give generalizations for the bounds defined in [8] over ω and α. In order to do that we need to present the following material.…”

“…, d(v n )) be the diagonal matrix of vertex degrees of G. The matrix L = D − A is known as the Laplacian matrix ( [3]) of G and has been studied extensively in the literature. Recently, there has also been interest in the matrix Q = D + A known as the signless Laplacian matrix ( [6,8,9]) of G. It is known that all eigenvalues of Q are non-negative since it is a positive semidefinite matrix. In the present paper, the eigenvalues of Q will be ordered by q 1 ≥ q 2 ≥ · · · ≥ q n ≥ 0, and the corresponding normalized eigenvectors will be represented by u 1 > 0, u 2 , .…”