  Let $G$ be a simple undirected $n$-vertex graph with the characteristic
polynomial of its Laplacian matrix $L(G)$, $\det (\lambda I - L (G))=\sum_{k =
0}^n (-1)^k c_k \lambda^{n - k}$. Laplacian--like energy of a graph is newly
proposed graph invariant, defined as the sum of square roots of Laplacian
eigenvalues. For bipartite graphs, the Laplacian--like energy coincides with
the recently defined incidence energy $IE (G)$ of a graph. In [D. Stevanovi\'
c, \textit{Laplacian--like energy of trees}, MATCH Commun. Math. Comput. Chem.
61 (2009), 407--417.] the author introduced a partial ordering of graphs based
on Laplacian coefficients. We point out that original proof was incorrect and
illustrate the error on the example using Laplacian Estrada index. Furthermore,
we found the inverse of Jacobian matrix with elements representing derivatives
of symmetric polynomials of order $n$, and provide a corrected elementary proof
of the fact: Let $G$ and $H$ be two $n$-vertex graphs; if for Laplacian
coefficients holds $c_k (G) \leqslant c_k (H)$ for $k = 1, 2, ..., n - 1$, then
$LEL (G) \leqslant LEL (H)$. In addition, we generalize this theorem and
provide a necessary condition for functions that satisfy partial ordering based
on Laplacian coefficients.
