Some observations on the Laplacian–energy–like invariant of trees

Let G be a graph of order n. Denote by A the adjacency matrix of G and by D = diag(d1, …, dn) the diagonal matrix of vertex degrees of G. The Laplacian matrix of G is defined as L = D − A. Let µ1, µ2, · · ·, µn−1, µn be eigenvalues of L satisfying (formula presented). The Laplacian-energy–like invariant is a graph invariant defined as (formula presented). Improved upper bounds for LEL(G) are obtained and compared when G has a tree structure.