Abstract: For a simple connected graph G, let A(G) and Q(G) be the adjacency matrix and signless Laplacian matrix, respectively of G. The principal eigenvector of A(G) (resp. Q(G)) is the unit positive eigenvector corresponding to the largest eigenvalue of A(G) (resp. Q(G)). In this paper, an upper bound and lower bound for the sum of the squares of the entries of the principal eigenvector of Q(G) corresponding to the vertices of an independent set are obtained.

Key words:  , Signless Laplacian, Principal eigenvector, Independent set, k-Regular, Semiregular