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

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