Normalized Laplacian eigenvalues and Randić energy of graphs

Let G be a connected graph of order n. Let d_i be the degree of the vertex v_i in G. The Randić matrix R = R(G) = (b_ij)_nxn is defined by b_ij = 1/√d_i d_j if the vertices v_i and v_j are adjacent, and b_ij = 0 otherwise. The Randić energy RE is the sum of absolute values of the eigenvalues of R. The normalized Laplacian matrix is defined as L(G) = I - R(G). We present necessary conditions under which the graphs G and G + {v_iv_j} have cospectral normalized Laplacian matrices and therefore equal Randić energies. In addition, we characterize some classes of graphs for which G and G + {v_iv_j} are not normalized-Laplacian cospectral. We also report results on Randić energy of edge-deleted cycles and paths.