Skew-spectra and skew energy of various products of graphs

‎‎Given a graph G‎, ‎let G^\sigma be an oriented graph of G with‎ ‎the orientation \sigma and skew-adjacency matrix S(G^\sigma)‎. ‎Then the spectrum of S(G^\sigma) consisting of all the eigenvalues of‎ ‎S(G^\sigma) is called the skew-spectrum of G^\sigma‎, ‎denoted by‎ ‎Sp(G^\sigma)‎. ‎The skew energy of the oriented graph G^\sigma‎, ‎denoted by \mathcal{E}_S(G^\sigma)‎, ‎is defined as the sum of the‎ ‎norms of all the eigenvalues of S(G^\sigma)‎. ‎In this paper‎, ‎we give orientations of the Kronecker product H\otimes G and the strong‎ ‎product H\ast G of H and G where H is a bipartite graph and G‎ ‎is an arbitrary graph‎. ‎Then we determine the skew-spectra of the resultant‎ ‎oriented graphs‎. ‎As applications‎, ‎we construct new families of oriented‎ ‎graphs with optimum skew energy‎. ‎Moreover‎, ‎we consider the skew energy of‎ ‎the orientation of the lexicographic product H[G] of a bipartite graph H‎ ‎and a graph G‎.

Keywords:

skew-spectrum , skew energy , Kronecker product , strong product , lexicographic product