Computing the permanental polynomials of bipartite graphs by Pfaffian orientation

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

Computing the permanental polynomial of a matrix from a combinatorial viewpoint

Recently, in the book [A Combinatorial Approach to Matrix Theory and Its Applications, CRC Press (2009)] the authors proposed a combinatorial approach to matrix theory by means of graph theory. In fact, if A is a square matrix over any field, then it is possible to associate to A a weighted digraph Ga, called Coates digraph. Through Ga (hence by graph theory) it is possible to express and prove results given for the matrix theory. In this paper we express the permanental polynomial of any matrix A in terms of permanental polynomials of some digraphs related to Ga