On the integral weighted oriented unicyclic graphs with minimum skew energy

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

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Detail(s)

Original languageEnglish
Pages (from-to)262-272
Journal / PublicationLinear Algebra and Its Applications
Volume439
Issue number1
Publication statusPublished - 1 Jun 2013

Abstract

Let Gσ be a weighted oriented graph, which is obtained from a simple weighted undirected graph by assigning an orientation to each of its edges. For a (weighted) oriented graph Gσ, the undirected graph obtained from Gσ by removing the orientation and the weight of each of its arcs is called the underlying graph of G σ, denoted by Ĝ. By U(n,m)(m≥n) we denote the set of all connected integral weighted oriented unicyclic graphs with order n in which each arc is assigned a positive integral weight and the sum of the weights of all arcs is equal to the specified integer m. In this paper, we investigate the minimal skew energies of integral weighted unicyclic oriented graphs, showing that the underlying graph of the oriented graph with minimum skew energy among all graphs over U(n,m)(n≥6) is Sn,3, the graph obtained from a triangle by attaching n-3 pendent edges in exactly one of its vertices. Moreover, we show that its weight sequence has form(w1,a, a,.,aï̧·k,a+ 1,a+1,.,a+1ï̧·n-3-k,w2,w3)in which the arc lying on the cycle C3 and incident to no pendent arcs has weight w1, and the two largest weights correspond the other two arcs of C3, where positive integer numbers w1,w 2,w3,a and k satisfy 1≤w1≤a,a+1≤w 2≤w3≤a+w1 and m=w1+ka+(n-k-3) (a+1)+w2+w3. In addition, we determine the weight sequence above for n≤m≤3n-1.

Research Area(s)

  • Integral weighted graph, Oriented graph, Skew energy, Skew symmetric matrix

Citation Format(s)

On the integral weighted oriented unicyclic graphs with minimum skew energy. / Gong, Shi-Cai; Hou, Yao-Ping; Woo, Ching-Wah; Xu, Guang-Hui; Shen, Xiao-Ling.

In: Linear Algebra and Its Applications, Vol. 439, No. 1, 01.06.2013, p. 262-272.

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review