Abstract

The Laplacian energy of a graph G is equal to the sum of distances of the Laplacian eigenvalues to their average, which in turn is equal to the sum of singular values of a shift of Laplacian matrix of G. Let X, Y, and Z be matrices, such that Z = X+Y. Ky Fan has established an inequality between the sum of singular values of Z and the sum of the sum of singular values of X and Y respectively. We apply this inequality to obtain new results in the theory of Laplacian energy of a graph.