New skew Laplacian energy of a simple digraph

TL;DR

This paper introduces a new skew Laplacian matrix and energy measure for simple directed graphs, providing bounds and characterizations of extremal digraphs.

Contribution

It proposes a novel skew Laplacian matrix for digraphs and derives bounds for its energy, expanding spectral graph theory tools for directed graphs.

Findings

Derived lower and upper bounds for the skew Laplacian energy.

Identified digraphs that attain these bounds.

Established properties of the new skew Laplacian matrix.

Abstract

For a simple digraph $G$ of order $n$ with vertex set $\{v_1,v_2,\ldots, v_n\}$, let $d_i^+$ and $d_i^-$ denote the out-degree and in-degree of a vertex $v_i$ in $G$, respectively. Let $D^+(G)=diag(d_1^+,d_2^+,\ldots,d_n^+)$ and $D^-(G)=diag(d_1^-,d_2^-,\ldots,d_n^-)$. In this paper we introduce $\widetilde{SL}(G)=\widetilde{D}(G)-S(G)$ to be a new kind of skew Laplacian matrix of $G$, where $\widetilde{D}(G)=D^+(G)-D^-(G)$ 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 $\widetilde{SL}(G)$. Some lower and upper bounds of the new skew Laplacian energy are derived and the digraphs attaining these bounds are also determined.