The skew-rank of oriented graphs

Original · EN

An oriented graph Gσ is a digraph without loops and multiple arcs, where G is called the underlying graph of Gσ. Let S(Gσ) denote the skew-adjacency matrix of Gσ. The rank of the skew-adjacency matrix of Gσ is called the skew-rank of Gσ, denoted by sr(Gσ). The skew-adjacency matrix of an oriented graph is skew symmetric and the skew-rank is even. In this paper we consider the skew-rank of simple oriented graphs. Firstly we give some preliminary results about the skew-rank. Secondly we characterize the oriented graphs with skew-rank 2 and characterize the oriented graphs with pendant vertices which attain the skew-rank 4. As a consequence, we list the oriented unicyclic graphs, the oriented bicyclic graphs with pendant vertices which attain the skew-rank 4. Moreover, we determine the skew-rank of oriented unicyclic graphs of order n with girth k in terms of matching number. We investigate the minimum value of the skew-rank among oriented unicyclic graphs of order n with girth k and characterize oriented unicyclic graphs attaining the minimum value. In addition, we consider oriented unicyclic graphs whose skew-adjacency matrices are nonsingular.