# 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 journal › peer-review

## Author(s)

## Related Research Unit(s)

## Detail(s)

Original language | English |
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Pages (from-to) | 262-272 |

Journal / Publication | Linear Algebra and Its Applications |

Volume | 439 |

Issue number | 1 |

Publication status | Published - 1 Jun 2013 |

## Link(s)

## Abstract

Let G

^{σ}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 G^{σ}, the undirected graph obtained from G^{σ}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 S_{n,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(w_{1},a,^{a,.,aï̧·k},a+^{1,a+1,.,a+1ï̧·n-3-k},w_{2},w_{3})in which the arc lying on the cycle^{C3}and incident to no pendent arcs has weight w_{1}, and the two largest weights correspond the other two arcs of^{C3}, where positive integer numbers w_{1},w_{2},w_{3},a and k satisfy 1≤w_{1}≤a,a+1≤w_{2}≤w_{3}≤a+w_{1}and m=w_{1}+ka+(n-k-3) (a+1)+w_{2}+w_{3}. 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 et al.

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 journal › peer-review