Abstract:

Let G be a simple undirected graph and G^\sigma the corresponding oriented graph of G with the orientation \sigma. G is said to be the underlying graph of G^\sigma. The skew Randi\'{c} matrix of an oriented graph G^\sigma is the real symmetric matrix R_{s}(G^\sigma)=[(r_s)_{ij}], where
(r_s)_{ij}=(d_id_j)^{-\frac{1}{2}} and (r_s)_{ji}=-(d_id_j)^{-\frac{1}{2}} if (v_i, v_j) is an arc of \sigma, otherwise (r_s)_{ij}=(r_s)_{ji}=0. The skew Randi\'{c} energy RE_s(G^\sigma) of G^\sigma is the sum of absolute values of the eigenvalues of R_{s}(G^\sigma). In this paper, we firstly
characterize the coefficients of the characteristic polynomial of R_{s}(G^\sigma). Secondly we give an integral representation for the skew Randi\'{c} energy of G^\sigma. Thirdly we show a new upper bound of RE_s(G^\sigma). Finally we compute RE_s(G^\sigma) of oriented cycles.

Key words: skew Randi\'{c} matrix, skew Randi\'{c} energy