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 Randi c matrix and the general Randi c index of a graph, we introduce the concept of general Randi c 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 Randi c eigenvalues are the eigenvalues of the general Randi c matrix, the greatest general Randi c eigenvalue is the general Randi c spectral radius of G, and the general Randi c energy is the sum of the absolute values of the general Randi c eigenvalues. In this paper, we prove some properties of the general Randi c matrix and obtain lower and upper bounds for general Randi c energy, also, we get some lower bounds for general Randi c spectral radius of a connected graph. Moreover, we give a new sharp upper bound for the general Randi c energy when = ??1=2.