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

Author

Listed:
  • Jing Huang

    (Central China Normal University)

  • Shuchao Li

    (Central China Normal University)

  • Hua Wang

    (Georgia Southern University)

Abstract

An oriented graph $$G^\sigma $$ G σ is a digraph without loops or multiple arcs whose underlying graph is G. Let $$S\left( G^\sigma \right) $$ S G σ be the skew-adjacency matrix of $$G^\sigma $$ G σ and $$\alpha (G)$$ α ( G ) be the independence number of G. The rank of $$S(G^\sigma )$$ S ( G σ ) is called the skew-rank of $$G^\sigma $$ G σ , denoted by $$sr(G^\sigma )$$ s r ( G σ ) . 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 $$sr(G^\sigma )+2\alpha (G)\geqslant 2|V_G|-2d(G)$$ s r ( G σ ) + 2 α ( G ) ⩾ 2 | V G | - 2 d ( G ) , where $$|V_G|$$ | 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 $$sr(G^\sigma )+\alpha (G),\, sr(G^\sigma )-\alpha (G)$$ s r ( G σ ) + α ( G ) , s r ( G σ ) - α ( G ) , $$sr(G^\sigma )/\alpha (G)$$ s r ( G σ ) / α ( G ) and characterize all corresponding extremal graphs.

Suggested Citation

  • Jing Huang & Shuchao Li & Hua Wang, 2018. "Relation between the skew-rank of an oriented graph and the independence number of its underlying graph," Journal of Combinatorial Optimization, Springer, vol. 36(1), pages 65-80, July.
  • Handle: RePEc:spr:jcomop:v:36:y:2018:i:1:d:10.1007_s10878-018-0282-x
    DOI: 10.1007/s10878-018-0282-x
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    References listed on IDEAS

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    1. Qu, Hui & Yu, Guihai, 2015. "Bicyclic oriented graphs with skew-rank 2 or 4," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 182-191.
    2. Lu, Yong & Wang, Ligong & Zhou, Qiannan, 2015. "Bicyclic oriented graphs with skew-rank 6," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 899-908.
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    Cited by:

    1. Jinling Yang & Ligong Wang & Xiuwen Yang, 2021. "Some mixed graphs with H-rank 4, 6 or 8," Journal of Combinatorial Optimization, Springer, vol. 41(3), pages 678-693, April.
    2. Feng, Zhimin & Huang, Jing & Li, Shuchao & Luo, Xiaobing, 2019. "Relationship between the rank and the matching number of a graph," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 411-421.
    3. Yong Lu & Ligong Wang & Qiannan Zhou, 2019. "The rank of a complex unit gain graph in terms of the rank of its underlying graph," Journal of Combinatorial Optimization, Springer, vol. 38(2), pages 570-588, August.
    4. Xueliang Li & Wen Xia, 2019. "Skew-rank of an oriented graph and independence number of its underlying graph," Journal of Combinatorial Optimization, Springer, vol. 38(1), pages 268-277, July.

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    4. Lu, Yong & Wang, Ligong & Zhou, Qiannan, 2015. "Bicyclic oriented graphs with skew-rank 6," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 899-908.

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