On the Difference Between the Skew-rank of an Oriented Graph and the Rank of Its Underlying Graph

Let G be a simple graph and G σ be the oriented graph with G as its underlying graph and orientation σ . The rank of the adjacency matrix of G is called the rank of G and is denoted by r ( G ). The rank of the skew-adjacency matrix of G σ is called the skew-rank of G σ and is denoted by sr ( G σ ). Let V ( G ) be the vertex set and E ( G ) be the edge set of G . The cyclomatic number of G , denoted by c ( G ), is equal to ∣ E ( G )∣ − ∣ V ( G )∣+ ω ( G ), where ω ( G ) is the number of the components of G . It is proved for any oriented graph G σ that −2 c ( G ) ⩽ sr( G σ ) − r ( G ) ⩽ 2 c ( G ). In this paper, we prove that there is no oriented graph G σ with sr ( G σ ) − r ( G ) = 2 c ( G )−1, and in addition, there are in nitely many oriented graphs G σ with connected underlying graphs such that c ( G ) = k and sr ( G σ )− r ( G ) = 2 c ( G )−ℓ for every integers k , ℓ satisfying 0 ⩽ ℓ ⩽ 4 k and ℓ≠ 1.