A decreasing sequence of upper bounds for the Laplacian energy of a tree

Let R be a nonnegative Hermitian matrix. The energy of R  , denoted by E(R)E(R), is the sum of absolute values of its eigenvalues. We construct an increasing sequence that converges to the Perron root of R  . This sequence yields a decreasing sequence of upper bounds for E(R)E(R). We then apply this result to the Laplacian energy of trees of order n  , namely to the sum of the absolute values of the eigenvalues of the Laplacian matrix, shifted by −2(n−1)/n−2(n−1)/n.