On Randic Energy

Let G be a simple graph with n vertices and m edges. Let d(i) be the degree of the i-th vertex of G. The Randic matrix R = (r(ij)) is defined by r(ij) = 1/root d(i)d(j) if the i-th and j-th vertices are adjacent and r(ij) = 0 otherwise. The Randic energy RE is the sum of absolute values of the eigenvalues of R. Cavers at al. [On the normalized Laplacian energy and general Randic index R-1(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 Delta, minimum degree delta, and the determinant of the adjacency matrix.