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

An oriented graph G ? is a digraph without loops and multiple arcs, where G is called the underlying graph of G ? . Let S ( G ? ) denote the skew-adjacency matrix of G ? , and A ( G ) be the adjacency matrix of G . The skew-rank of G ? , written as s r ( G ? ) , refers to the rank of S ( G ? ) , which is always even since S ( G ? ) is skew symmetric.A natural problem is: How about the relation between the skew-rank of an oriented graph G ? and the rank of its underlying graph? In this paper, we focus our attention on this problem. Denote by d ( G ) the dimension of cycle spaces of G , that is d ( G ) = | E ( G ) | - | V ( G ) | + ? ( G ) , where ? ( G ) denotes the number of connected components of G . It is proved that s r ( G ? ) ? r ( G ) + 2 d ( G ) for an oriented graph G ? , the oriented graphs G ? whose skew-rank attains the upper bound are characterized.

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