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 ( d 1 , . . . , d n ) the diagonal matrix of vertex degrees of G . The Laplacian matrix of G is defined as L = D − A . Let µ 1 , µ 2 , · · · , µ n − 1 , µ n be eigenvalues of L satisfying µ 1 ≥ µ 2 ≥ · · · ≥ µ n − 1 ≥ µ n = 0 . The Laplacian-energy–like invariant is a graph invariant defined as LEL ( G ) = (cid:80) n − 1 i =1 √ µ i . Improved upper bounds for LEL ( G ) are obtained and compared when G has a tree structure.