Ky Fan theorem applied to Randić energy

Let G be a simple undirected graph of order n   with vertex set V(G)={v1,v2,…,vn}V(G)={v1,v2,…,vn}. Let didi be the degree of the vertex vivi. The Randić matrix R=(ri,j)R=(ri,j) of G is the square matrix of order n   whose (i,j)(i,j)-entry is equal to 1/didj if the vertices vivi and vjvj are adjacent, and zero otherwise. The Randić energy is the sum of the absolute values of the eigenvalues of R. Let X, Y, and Z be matrices, such that X+Y=ZX+Y=Z. Ky Fan established an inequality between the sum of singular values of X, Y, and Z. We apply this inequality to obtain bounds on Randić energy. We also present results pertaining to the energy of a symmetric partitioned matrix, as well as an application to the coalescence of graphs.