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 LiEmail author
  • Hua Wang


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.


Skew-rank Oriented graph Evenly-oriented Independence number 

Mathematics Subject Classification




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.


  1. Anuradha A, Balakrishnan R (2013) Skew spectrum of the Cartesian product of an oriented graph with an oriented Hypercube. In: Bapat RB, Kirkland SJ, Prasad KM, Puntanen S (eds) Combinatorial matrix Theory and generalized inverses of matrices. Springer, New York, pp 1–12Google Scholar
  2. Anuradha A, Balakrishnan R, Chen X, Li X, Lian H, So W (2013) Skew spectra of oriented bipartite graphs. Electron J Comb 20(4):P19MathSciNetzbMATHGoogle Scholar
  3. Aouchiche M (2006) Comparaison automatisée d’invariants en théorie des graphes, Ph.D. Thesis, École Polytechnique de MontréalGoogle Scholar
  4. Aouchiche M, Bonnefoy JM, Fidahoussen A, Caporossi G, Hansen P, Hiesse L, Lacher J, Monhait A (2005) Variable neighborhood search for extremal graphs. 14. The AutoGraphiX 2 system. In: Liberti L, Maculan N (eds) Global optimization: from theory to implementation. Springer, New YorkGoogle Scholar
  5. Bondy JA, Murty USR (2008) Graph theory. In: Axler S, Ribet KA (eds) Graduate texts in mathematics, vol 244. SpringerGoogle Scholar
  6. Caporossi G, Hansen P (2000) Variable neighborhood search for extremal graphs. 1. The AutoGraphiX system. Discrete Math 212:29–44MathSciNetCrossRefzbMATHGoogle Scholar
  7. Caporossi G, Hansen P (2004) Variable neighborhood search for extremal graphs. 5. Three ways to automate finding conjectures. Discrete Math 276:81–94MathSciNetCrossRefzbMATHGoogle Scholar
  8. Cavers M, Cioabǎ SM, Fallat S, Gregory DA, Haemers WH, Kirkland SJ, McDonald JJ, Tsatsomeros M (2012) Skew-adjacency matrices of graphs. Linear Algebra Appl 436:4512–4529MathSciNetCrossRefzbMATHGoogle Scholar
  9. Cvetković D, Doob M, Sachs H (1995) Spectra of graphs, 3rd edn. Johann Ambrosius Barth, HeidelbergzbMATHGoogle Scholar
  10. Haranta J, Schiermeyer I (2001) Note on the independence number of a graph in terms of order and size. Discrete Math 232:131–138MathSciNetCrossRefGoogle Scholar
  11. Hou YP, Lei T (2011) Charactristic polynomials of skew-adjacency matrices of oriented graphs. Electron J Comb 18:P156zbMATHGoogle Scholar
  12. Huang J, Li SC (under review) Further relation between the skew-rank of an oriented graph and the rank of its underlying graphGoogle Scholar
  13. IMA-IS Research Group on Minimum Rank (2010) Minimum rank of skew-symmetric matrices described by a graph. Linear Algebra Appl 432:2457–2472MathSciNetCrossRefGoogle Scholar
  14. Li XL, Yu GH (2015) The skew-rank of oriented graphs. Sci Sin Math 45:93–104 (Chinese)Google Scholar
  15. Lu Y, Wang LG, Zhou QN (2015) Bicyclic oriented graphs with skew-rank 6. Appl Math Comput 270:899–908MathSciNetGoogle Scholar
  16. Ma XB, Wong DY, Tian FL (2016a) Skew-rank of an oriented graph in terms of matching number. Linear Algebra Appl 495:242–255MathSciNetCrossRefzbMATHGoogle Scholar
  17. Ma XB, Wong DY, Tian FL (2016b) Nullity of a graph in terms of the dimension of cycle space and the number of pendant vertices. Discrete Appl Math 215:171–176MathSciNetCrossRefzbMATHGoogle Scholar
  18. Mallik S, Shader BL (2013) Classes of graphs with minimum skew rank 4. Linear Algebra Appl 439:3643–3657MathSciNetCrossRefzbMATHGoogle Scholar
  19. Mallik S, Shader BL (2016) On graphs of minimum skew rank 4. Linear Multilinear Algebra 64:279–289MathSciNetCrossRefzbMATHGoogle Scholar
  20. Pumplün S (2010) Classes of structurable algebras of skew-rank 1. Isr J Math 180:425–460MathSciNetCrossRefzbMATHGoogle Scholar
  21. Pumplün S (2011) Structurable algebras of skew-rank 1 over the affine plane. Pac J Math 254:361–380MathSciNetCrossRefzbMATHGoogle Scholar
  22. Qu H, Yu GH (2015) Bicyclic oriented graphs with skew-rank 2 or 4. Appl Math Comput 258:182–191MathSciNetzbMATHGoogle Scholar
  23. Qu H, Yu GH, Feng LH (2015) More on the minimum skew-rank of graphs. Oper Matrices 9:311–324MathSciNetCrossRefzbMATHGoogle Scholar
  24. Shader B, So WS (2009) Skew spectra of oriented graphs. Electron J Comb 16(Note 32):6MathSciNetzbMATHGoogle Scholar
  25. Wong DY, Ma XB, Tian FL (2016) Relation between the skew-rank of an oriented graph and the rank of its underlying graph. Eur J Comb 54:76–86MathSciNetCrossRefzbMATHGoogle Scholar
  26. Xu GH (2012) Some inequlities on the skew-spectral radii of oriented graphs. J Inequal Appl 2012:211CrossRefzbMATHGoogle Scholar

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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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