on laplacian-energy-like invariant and incidence energy

For a simple connected graph g with n-vertices having laplacian eigenvalues µ1, µ2, . . . , µn−1, µn = 0, and signless laplacian eigenvalues q1, q2, . . . , qn, the laplacian-energy-like invariant (lel) and the incidence energy (ie) of a graph g are respectively defined as lel(g) = ∑n−1 i=1 √µi and ie(g) = ∑n i=1 √qi. in this paper, we obtain some sharp lower and upper bounds for the laplacian-energy-like invariant and incidence energy of a graph.