On the Permanental Polynomials of Some Graphs |
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Authors: | Weigen Yan Fuji Zhang |
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Affiliation: | (1) Department of Mathematics, Jimei University, Xiamen, 361021, P.R. China;(2) Department of Mathematics, Xiamen University, Xiamen, 361005, P.R. China |
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Abstract: | Let G be a simple graph with adjacency matrix A(G) and (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 (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
with 2n+2 vertices such that (G,x) is a factor of
.3.Let T be an arbitrary tree with n vertices. Then
, where
1
,
2
, ...,
n are the eigenvalues of T. |
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Keywords: | outerplanar graph adjacency matrix skew adjacency matrix characteristic polynomial permanental polynomial Cartesian product Pfaffian orientation nice cycle |
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