On the permanental polynomials of some graphs∗

Let G be a simple graph with adjacency matrix A(G) and π(G,x) the permanental polynomial of G. Let G × H denotes the Cartesian product of graphs G and H. Inspired by Klein’s idea to compute the permanent of some matrices (Mol. Phy., 1976, Vol. 31, (3): 811−823), in this paper in terms of some orientation of graphs we study the permanental polynomial of a type of graphs. Here are some of our main results. 1 If G is a bipartite graph containing no subgraph which is an even subdivision of K2,3, then G has an orientation G e such that π(G,x) = det(xI − A(G)), where A(G) denotes the skew adjacency matrix of G. 2 Let G be a 2−connected outerplanar bipartite graph with n vertices. Then there exists a 2−connected outerplanar bipartite graph G with 2n + 2 vertices such that π(G,x) is a factor of π(G,x).