Bounds for the skew Laplacian energy of weighted digraphs

• Published in: Afrika mathematica 2021-09, Vol.32 (5-6), p.745-756
• Main Authors: Chat, Bilal A., Ganie, Hilal A., Pirzada, S.
• Format: Article
• Language: English
• Subjects: Apexes; Applications of Mathematics; Eigenvalues; Graph theory; History of Mathematical Sciences; Mathematics; Mathematics and Statistics; Mathematics Education; Real numbers; Upper bounds
• ISSN: 1012-9405; 2190-7668
• DOI: 10.1007/s13370-020-00858-2

Let D be a simple connected digraph with n vertices and m arcs and let W ( D ) = ( D , ω ) be the weighted digraph corresponding to D , where the weights are taken from the set of non-zero real numbers. In this paper, we define the skew Laplacian matrix S L ( W ( D ) ) and skew Laplacian energy S L E ( W ( D ) ) of a weighted digraph W ( D ) , which is defined as the sum of the absolute values of the skew Laplacian eigenvalues, that is, S L E ( W ( D ) ) = ∑ i = 1 n | ρ i | , where ρ 1 , ρ 2 , … , ρ n are the skew Laplacian eigenvalues of W ( D ) . We show the existence of the real skew Laplacian eigenvalues of a weighted digraph when the weighted digraph has an independent set such that all the vertices in the independent set have the same out-neighbors and in-neighbors. We obtain a Koolen type upper bound for S L E ( W ( D ) ) . Further, for a connected weighted digraph W ( D ) , we obtain bounds for S L E ( W ( D ) ) , in terms of different digraph parameters associated with the digraph structure D . We characterize the extremal weighted digraphs attaining these bounds.