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

Abstract

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.