Randic matrix and randić energy

If G is a graph on n vertices, and di, is the degree of its i-th vertex, then the Randiá matrix of G is the square matrix of order n whose (i, j)-entry is equal to √didi if the i-th and j-th vertex of G are adjacent, and zero otherwise. This matrix in a natural way occurs within Laplacian spectral theory, and provides the non-trivial part of the so-called normalized Laplacian matrix. In spite of its obvious relation to the famous Randić index, the Randić matrix seems to have not been much studied in mathematical chemistry. In this paper we define the Randić energy as the sum of the absolute values of the eigenvalues of the Randić matrix, and establish some of its properties, in particular lower and upper bounds for it.