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Journal of Combinatorial Optimization

, Volume 36, Issue 1, pp 65–80 | Cite as

Relation between the skew-rank of an oriented graph and the independence number of its underlying graph

  • Jing Huang
  • Shuchao Li
  • Hua Wang
Article

Abstract

An oriented graph \(G^\sigma \) is a digraph without loops or multiple arcs whose underlying graph is G. Let \(S\left( G^\sigma \right) \) be the skew-adjacency matrix of \(G^\sigma \) and \(\alpha (G)\) be the independence number of G. The rank of \(S(G^\sigma )\) is called the skew-rank of \(G^\sigma \), denoted by \(sr(G^\sigma )\). Wong et al. (Eur J Comb 54:76–86, 2016) studied the relationship between the skew-rank of an oriented graph and the rank of its underlying graph. In this paper, the correlation involving the skew-rank, the independence number, and some other parameters are considered. First we show that \(sr(G^\sigma )+2\alpha (G)\geqslant 2|V_G|-2d(G)\), where \(|V_G|\) is the order of G and d(G) is the dimension of cycle space of G. We also obtain sharp lower bounds for \(sr(G^\sigma )+\alpha (G),\, sr(G^\sigma )-\alpha (G)\), \(sr(G^\sigma )/\alpha (G)\) and characterize all corresponding extremal graphs.

Keywords

Skew-rank Oriented graph Evenly-oriented Independence number 

Mathematics Subject Classification

05C50 

Notes

Acknowledgements

The authors would like to express their sincere gratitude to all of the referees for their insightful comments and suggestions, which led to a number of improvements to this paper.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and StatisticsCentral China Normal UniversityWuhanPeople’s Republic of China
  2. 2.Department of Mathematical SciencesGeorgia Southern UniversityStatesboroUSA

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