A sharp upper bound on the maximal entry in the principal eigenvector of symmetric nonnegative matrix

An upper bound on the maximal entry in the principal eigenvector of a symmetric nonnegative matrix with zero diagonal entries is investigated in [S. Zhao, Y. Hong, On the bounds of maximal entries in the principal eigenvector of symmetric nonnegative matrix, Linear Algebra Appl. 340 (2002) 245–252]. We obtain a sharp upper bound on the maximal entry ymaxp in the principal eigenvector of symmetric nonnegative matrix in terms of order, the spectral radius, the largest and the smallest diagonal entries of that matrix. Our bound is applicable for any symmetric nonnegative matrix and the upper bound of Zhao and Hong (2002) for the maximal entry ymaxp follows as a special case. Moreover, we find an upper bound on maximal entry in the principal eigenvector for the signless Laplacian matrix of a graph.