Applications of a theorem by Ky Fan in the theory of weighted Laplacian graph energy

The energy of a graph $G$ is equal to the sum of the absolute values of the eigenvalues of $G$ , which in turn is equal to the sum of the singular values of the adjacency matrix of $G$. Let $X$, $Y$ and $Z$ be matrices, such that $X+Y= Z$. The Ky Fan theorem establishes an inequality between the sum of the singular values of $Z$ and the sum of the sum of the singular values of $X$ and $Y$. This theorem is applied in the theory of graph energy, resulting in several new inequalities, as well as new proofs of some earlier known inequalities.