Computing the permanental polynomials of bipartite graphs by Pfaffian orientation

The permanental polynomial of a graph GG is π(G,x)≜per(xI−A(G)). From the result that a bipartite graph GG admits an orientation GeGe such that every cycle is oddly oriented if and only if it contains no even subdivision of K2,3K2,3, Yan and Zhang showed that the permanental polynomial of such a bipartite graph GG can be expressed as the characteristic polynomial of the skew adjacency matrix A(Ge)A(Ge). In this note we first prove that this equality holds only if the bipartite graph GG contains no even subdivision of K2,3K2,3. Then we prove that such bipartite graphs are planar. Unexpectedly, we mainly show that a 2-connected bipartite graph contains no even subdivision of K2,3K2,3 if and only if it is planar 1-cycle resonant. This implies that each cycle is oddly oriented in any Pfaffian orientation of a 2-connected bipartite graph containing no even subdivision of K2,3K2,3. Accordingly, we give a way to compute the permanental polynomials of such graphs by Pfaffian orientation.