Skew Spectrum of the Cartesian Product of an Oriented Graph with an Oriented Hypercube

Abstract

Let σ be an orientation of a simple graph H yielding an oriented graph H ^σ. We define an orientation ψ to the Cartesian product G=H□Q _d of H with the hypercube Q _d by orienting the edges of G in a specific way. The skew adjacency matrices S(G ^ψ) obtained in this way for some special families of G answer some special cases of the Inverse Eigenvalue Problem. Further we present a new orientation ϕ to the hypercube Q _d for which the skew energy equals the energy of the underlying hypercube, distinct from the two orientations of hypercubes defined by Tian (Linear Algebra Appl. 435:2140–2149, 2011) and show how one of the two orientations of Q _d described by Tian is a special case of our method.