On the coefficients of skew Laplacian characteristic polynomial of digraphs

Abstract

Let [Formula: see text] be a connected graph with [Formula: see text] vertices and [Formula: see text] edges. Let [Formula: see text] be the digraph obtained by orienting the edges of [Formula: see text] arbitrarily. The digraph [Formula: see text] is called an orientation of [Formula: see text] or oriented graph corresponding to [Formula: see text]. The skew Laplacian matrix of the digraph [Formula: see text] is denoted by [Formula: see text] and is defined as [Formula: see text], where [Formula: see text] is the skew matrix and [Formula: see text] is the diagonal matrix with [Formula: see text]th diagonal entry [Formula: see text]. In this paper, we obtain combinatorial representation for the first five coefficients of characteristic polynomial of skew Laplacian matrix of [Formula: see text]. 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.