GENERAL RANDIC MATRIX AND GENERAL RANDIC ENERGY

Let G be a simple graph with vertex set V (G) = fv1; v2; : : : ; vng and di the degree of its vertex vi, i = 1; 2; : : : ; n. Inspired by the Randic matrix and the general Randic index of a graph, we introduce the concept of general Randic matrix R of G, which is dened by (R)i;j = (didj) if vi and vj are adjacent, and zero otherwise. Similarly, the general Randic eigenvalues are the eigenvalues of the general Randic matrix, the greatest general Randic eigenvalue is the general Randic spectral radius of G, and the general Randic energy is the sum of the absolute values of the general Randic eigenvalues. In this paper, we prove some properties of the general Randic matrix and obtain lower and upper bounds for general Randic energy, also, we get some lower bounds for general Randic spectral radius of a connected graph. Moreover, we give a new sharp upper bound for the general Randic energy when  = ??1=2.