On randić energy

Let G be a simple graph with n vertices and m edges. Let di be the degree of the i-th vertex of G. The Randić matrix R = (r_ij) is defined by r_ij = 1/√d_id_j if the i-th and j-th vertices are adjacent and r_ij = 0 otherwise. The Randić energy RE is the sum of absolute values of the eigenvalues of R. Cavers et al. [On the normalized Laplacian energy and general Randić index R₁(G) of graphs, Lin. Algebra Appl. 433 (2010) 172-190] obtained some bounds on RE, but did not characterize the extremal graphs. We now find these extremal graphs. Additional lower and upper bounds for RE are obtained, in terms of n, m, maximum degree Δ, minimum degree δ, and the determinant of the adjacency matrix.