Geodesic
The Graphs Whose Permanental Polynomials Are Symmetric
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 1, pp. 233-243.

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The permanental polynomial π (G,x) = Σ_i=0^n b_i x^n − i of a graph G is symmetric if b_i = b_n−i for each i. In this paper, we characterize the graphs with symmetric permanental polynomials. Firstly, we introduce the rooted product H(K) of a graph H by a graph K, and provide a way to compute the permanental polynomial of the rooted product H(K). Then we give a sufficient and necessary condition for the symmetric polynomial, and we prove that the permanental polynomial of a graph G is symmetric if and only if G is the rooted product of a graph by a path of length one.
Keywords: permanental polynomial, rooted product, matching 
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Li, Wei. The Graphs Whose Permanental Polynomials Are Symmetric. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 1, pp. 233-243. http://geodesic.mathdoc.fr/item/DMGT_2018_38_1_a18/

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