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Variable Neighborhood Search for Extremal Graphs 14: The AutoGraphiX 2 System
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.
Journal of Combinatorial Optimization – Springer Journals
Published: Mar 30, 2018
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