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We consider the skew Laplacian matrix of a digraph [Formula: see text] obtained by giving an arbitrary direction to the edges of a graph [Formula: see text] having [Formula: see text] vertices and [Formula: see text] edges. With [Formula: see text] to be the skew Laplacian eigenvalues of [Formula: see text], the skew Laplacian energy [Formula: see text] of [Formula: see text] is defined as [Formula: see text]. In this paper, we analyze the effect of changing the orientation of an induced subdigraph on the skew Laplacian spectrum. We obtain bounds for the skew Laplacian energy [Formula: see text] in terms of various parameters associated with the digraph [Formula: see text] and the underlying graph [Formula: see text] and we characterize the extremal digraphs attaining these bounds. We also show these bounds improve some known bounds for some families of digraphs. Further, we show the existence of some families of skew Laplacian equienergetic digraphs.

We consider the skew Laplacian matrix of a digraph −→G obtained by giving an arbitrary direction to the edges of a graph G having n vertices and m edges. We obtain an upper bound for the skew Laplacian spectral radius in terms of the adjacency and the signless Laplacian spectral radius of the underlying graph G. We also obtain upper bounds for the skew Laplacian spectral radius and skew spectral radius, in terms of various parameters associated with the structure of the digraph −→G and characterize the extremal graphs.

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The subdivision graph S(G) of a graph G is the graph obtained by inserting a new vertex into every edge of G. The Svertex or Sver join of the graph G1 with the graph G2, denoted by G1∨G2, is obtained from S(G1) and G2 by joining all vertices of G1 with all vertices of G2. The S edge or S ed join of G1 and G2, denoted by G1∨G2, is obtained from S(G1) and G2 by joining all vertices of S(G1) corresponding to the edges of G1 with all vertices of G2. In this paper, we obtain graphical sequences of the family of induced subgraphs of SJ = G1 ∨ G2, Sver = G1∨G2 and S ed = G1∨G2. Also we prove that the graphic sequence of S ed is potentially K4 − e-graphical. MSC: 05C07

For a simple connected graph G with n vertices and m edges, let − → G be a digraph obtained by giving an arbitrary direction to the edges of G . In this paper, we consider the skew Laplacian matrix of a digraph − → G and we obtain the skew Laplacian spectrum of the orientations of a complete bipartite graph, complete split graph and the join of two graphs. We prove that deleting an edge of a Hamiltonian path in a transitive tournament does not effect the skew Laplacian spectrum. We show the existence of various families of skew Laplacian integral digraphs.

Let [Formula: see text] be an orientation of a simple graph [Formula: see text] with [Formula: see text] vertices and [Formula: see text] edges. The skew Laplacian matrix [Formula: see text] of the digraph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the imaginary unit, [Formula: see text] is the diagonal matrix with oriented degrees [Formula: see text] as diagonal entries and [Formula: see text] is the skew matrix of the digraph [Formula: see text]. The largest eigenvalue of the matrix [Formula: see text] is called skew Laplacian spectral radius of the digraph [Formula: see text]. In this paper, we study the skew Laplacian spectral radius of the digraph [Formula: see text]. We obtain some sharp lower and upper bounds for the skew Laplacian spectral radius of a digraph [Formula: see text], 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.

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