On the skew-permanental polynomials of orientation graphs

Let π(G,x)=∑i=0nbixn−i be the permanental polynomial of a graph GG, and πs(Gσ,x)=∑i=0ndixn−i the skew-permanental polynomial of an orientation graph GσGσ. In this paper, we investigate first the orientation graph GσGσ of a bipartite graph GG satisfying |dn|=|bn||dn|=|bn|. Furthermore, we characterize the orientation of a bipartite graph with |di|=|bi||di|=|bi| for each ii. Then we establish the recursion formulas for the skew-permanental polynomials of orientation graphs. In addition, we characterize the graphs whose skew-permanental polynomials can be expressed by the corresponding characteristic polynomials.