On the signless Laplacian spectral radius of irregular graphs

Abstract Let G be a connected irregular graph. In this paper, we give lower and upper bounds for the signless Laplacian radius q 1 of G . We show that q 1 > 4 e n + ( Δ - δ ) 2 2 n Δ and q 1 2 Δ - 1 n ( d - 1 4 ) hold, where d is the diameter of G .