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Skew-rank of an oriented graph in terms of the rank and dimension of cycle space of its underlying graph

Lu Yong (Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi’an, Shaanxi, P.R. China)
Wang Ligong (Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi’an, Shaanxi, P.R. China)
Zhou Qiannan (Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi’an, Shaanxi, P.R. China)

Let Gσ be an oriented graph and S(Gσ) be its skew-adjacency matrix, where G is called the underlying graph of Gσ. The skew-rank of Gσ, denoted by sr(Gσ), is the rank of S(Gσ). Denote by d(G) = |E(G)|-|V(G)| + θ(G) the dimension of cycle spaces of G, where |E(G)|, |V(G)| and θ(G) are the edge number, vertex number and the number of connected components of G, respectively. Recently, Wong, Ma and Tian [European J. Combin. 54 (2016) 76-86] proved that sr(Gσ) ≤ r(G) + 2d(G) for an oriented graph Gσ, where r(G) is the rank of the adjacency matrix of G, and characterized the graphs whose skew-rank attain the upper bound. However, the problem of the lower bound of sr(Gσ) of an oriented graph Gσ in terms of r(G) and d(G) of its underlying graph G is left open till now. In this paper, we prove that sr(Gσ) ≥ r(G)-2d(G) for an oriented graph Gσ and characterize the graphs whose skew-rank attain the lower bound.

Keywords: Skew-rank, Rank of graphs, Dimension of cycle space