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

Abstract

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 −2c(G) ⩽ sr(G^σ) − r(G) ⩽ 2c(G). In this paper, we prove that there is no oriented graph G^σ with sr(G^σ) − r(G) = 2c(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) = 2c(G)−ℓ for every integers k, ℓ satisfying 0 ⩽ ℓ ⩽ 4k and ℓ≠ 1.