New skew Laplacian energy of a simple digraph

الأصل · EN

For a simple digraph G of order n with vertex set {v₁,v₂,, vₙ}, let dᵢ+ and dᵢ- denote the out-degree and in-degree of a vertex vᵢ in G, respectively. Let D+(G)=diag(d₁+,d₂+,,dₙ+) and D-(G)=diag(d₁-,d₂-,,dₙ-). In this paper we introduce SL(G)=D(G)-S(G) to be a new kind of skew Laplacian matrix of G, where D(G)=D+(G)-D-(G) and S(G) is the skew-adjacency matrix of G, and from which we define the skew Laplacian energy SLE(G) of G as the sum of the norms of all the eigenvalues of SL(G). Some lower and upper bounds of the new skew Laplacian energy are derived and the digraphs attaining these bounds are also determined.