A Ky Fan theorem’s application in the theory of graph energy

The energy of a graph G is equal to the total of its absolute eigenvalues, xy,zwhich is equal to the sum of its singular adjacency matrix values. Let and be x+=y zmatrices such that . The Ky-Fan theorem proves an inequality between the yxzsum of the singular values of and the sum of the singular values of and . Several new inequalities as well as fresh proofs of several previously known inequalities are produced when this theorem is applied to the notion of graph energy.