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

## 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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