New skew Laplacian energy of a simple digraph

2013/04/24 by Qingqiong Cai, Xueliang Li, Cai, Qingqiong +3
Mathematics · #05C20 #05C50 #05C90 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C20 #msc:05C50 #msc:05C90

paper · pdf · doi:10.48550/arxiv.1304.6465

13 pages

arxiv created 2013/04/24 · arxiv updated 2013/04/25

Abstract

For a simple digraph G of order n with vertex set \v₁,v₂,…, 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(d₁⁺,d₂⁺,…,d_n⁺) and D^-(G)=diag(d₁^-,d₂^-,…,d_n^-). In this paper we introduce \widetildeSL(G)=\widetildeD(G)-S(G) to be a new kind of skew Laplacian matrix of G, where \widetildeD(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 \widetildeSL(G). Some lower and upper bounds of the new skew Laplacian energy are derived and the digraphs attaining these bounds are also determined.

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