On the skew energy of orientations of hypercubes

Abstract

Let D be a digraph of order n and λ 1 , λ 2 , … , λ n denote all the eigenvalues of the skew-adjacency matrix of D. The skew energy E S ( D ) of D is defined as E S ( D ) = ∑ i = 1 n | λ i | . In this paper, it is proved that for any positive integer k ≥ 3 , there exists a k-regular graph of order n having an orientation D with E S ( D ) = n k . This work positively answers a problem proposed by Adiga et al. [C. Adiga, R. Balakrishnan, Wasin So, The skew energy of a digraph, Linear Algebra Appl. 432 (2010) 1825–1835]. In addition, a digraph is also constructed such that its skew energy is the same as the energy of its underlying graph.