On the coefficients of skew Laplacian characteristic polynomial of digraphs.

Let G be a connected graph with n vertices and m edges. Let G ⃗ be the digraph obtained by orienting the edges of G arbitrarily. The digraph G ⃗ is called an orientation of G or oriented graph corresponding to G. The skew Laplacian matrix of the digraph G ⃗ is denoted by S L ̃ (G ⃗) and is defined as S L ̃ (G ⃗) = D ̃ (G ⃗) − S (G ⃗) , where S (G ⃗) is the skew matrix and D ̃ (G ⃗) is the diagonal matrix with i th 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 ⃗. 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. [ABSTRACT FROM AUTHOR]