On the Permanental Polynomials of Some Graphs

Abstract

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. 31 (3) (1976) 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 K _2,3, then G has an orientation G ^e such that π(G,x) = det (xI-A(G ^e )), where A(G ^e ) denotes the skew adjacency matrix of G ^e.

2.Let G be a 2-connected outerplanar bipartite graph with n vertices. Then there exists a 2-connected outerplanar bipartite graph $\overline G$ with 2n+2 vertices such that π(G,x) is a factor of $\pi (\overline G ,x)$.

3.Let T be an arbitrary tree with n vertices. Then $\pi (T \times K_2 ,x)\prod _{i = 1}^n (x^2 + 1 + \alpha _i^2 )$, where α ₁ , α ₂ , ..., α _n are the eigenvalues of T.