General Randic matrix and general Randic energy

Let $G$ be a simple graph with vertex set $V(G) = {v_1, v_2,ldots, v_n}$ and $d_i$ the degree of its vertex $v_i$, $i = 1, 2, cdots, n$. Inspired by the Randic matrix and the general Randic index of a graph, we introduce the concept of general Randic matrix $textbf{R}_alpha$ of $G$, which is defined by $(textbf{R}_alpha)_{i,j}=(d_id_j)^alpha$ if $v_i$ and $v_j$ 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 Randic 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 $alpha=-1/2$.[2mm] noindent{bf Keywords:} general Randic matrix, general Randic energy, eigenvalues, spectral radius.