On the skew Laplacian spectral radius of a digraph

Let G be an orientation of a simple graph G with n vertices and m edges. The skew Laplacian matrix SL(G) of the digraph G is defined as SL(G) = (D) over tilde (G) - iS(G), where i = -1 is the imaginary unit, (D) over tilde (G) is the diagonal matrix with oriented degrees alpha(i) = d(i)(+) - d(i)(-) as diagonal entries and S(G) is the skew matrix of the digraph G. The largest eigenvalue of the matrix SL(G) is called skew Laplacian spectral radius of the digraph G. In this paper, we study the skew Laplacian spectral radius of the digraph G. We obtain some sharp lower and upper bounds for the skew Laplacian spectral radius of a digraph G, in terms of different structural parameters of the digraph and the underlying graph. We characterize the extremal digraphs attaining these bounds in some cases. Further, we end the paper with some problems for the future research in this direction.