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 sigma be the oriented graph with G as its underlying graph and orientation sigma. 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 sigma is called the skew-rank of G sigma and is denoted by sr(G sigma). 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 divide E(G) divide - divide V(G) divide + omega(G), where omega(G) is the number of the components of G. It is proved for any oriented graph G sigma that -2c(G) <= sr(G sigma) - r(G) <= 2c(G). In this paper, we prove that there is no oriented graph G sigma with sr(G sigma) - r(G) = 2c(G)-1, and in addition, there are in nitely many oriented graphs G sigma with connected underlying graphs such that c(G) = k and sr(G sigma)-r(G) = 2c(G)-l for every integers k, l satisfying 0 <= l <= 4k and l not equal 1.