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 randi'c matrix and the general randi'cindex of a graph, we introduce the concept of general randi'cmatrix $textbf{r}_alpha$ of $g$, which is defined by$(textbf{r}_alpha)_{i,j}=(d_id_j)^alpha$ if $v_i$ and $v_j$ areadjacent, 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 sumof the absolute values of the general randi'{c} eigenvalues. inthis paper, we prove some properties of the general randi'c matrixand obtain lower and upper bounds for general randi'{c} energy,also, we get some lower bounds for general randi'{c} spectralradius of a connected graph. moreover, we give a new sharp upperbound for the general randi'{c} energy when $alpha=-1/2$.