On the coefficients of skew Laplacian characteristic polynomial of digraphs

Let G be a connected graph with n vertices and m edges. Let (G) over right arrow be the digraph obtained by orienting the edges of G arbitrarily. The digraph (G) over right arrow is called an orientation of G or oriented graph corresponding to G. The skew Laplacian matrix of the digraph (G) over right arrow is denoted by (SL) over tilde((G) over right arrow) and is defined as (SL) over tilde((G) over right arrow) = (D) over tilde((G) over right arrow) - S((G) over right arrow), where S((G) over right arrow) is the skew matrix and (D) over tilde((G) over right arrow) is the diagonal matrix with ith diagonal entry d(i)(+)- d(i)(-). In this paper, we obtain combinatorial representation for the first five coefficients of characteristic polynomial of skew Laplacian matrix of (G) over right arrow. We provide examples of orientations of some well-known graphs to highlight the importance of our results. We conclude the paper with some observations about the skew Laplacian spectral determinations of the directed path and directed cycle.