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A Note on the Permanental Roots of Bipartite Graphs

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Zhang H., Liu S., Li W.
Discussiones Mathematicae Graph Theory
|
2014
|
tom 34
|
nr 1
49-56
EN
It is well-known that any graph has all real eigenvalues and a graph is bipartite if and only if its spectrum is symmetric with respect to the origin. We are interested in finding whether the permanental roots of a bipartite graph G have symmetric property as the spectrum of G. In this note, we show that the permanental roots of bipartite graphs are symmetric with respect to the real and imaginary axes. Furthermore, we prove that any graph has no negative real permanental root, and any graph containing at least one edge has complex permanental roots.
2
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Per-Spectral Characterizations Of Some Bipartite Graphs

100%
Wu T., Zhang H.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 4
935-951
EN
A graph is said to be characterized by its permanental spectrum if there is no other non-isomorphic graph with the same permanental spectrum. In this paper, we investigate when a complete bipartite graph Kp,p with some edges deleted is determined by its permanental spectrum. We first prove that a graph obtained from Kp,p by deleting all edges of a star K1,l, provided l < p, is determined by its permanental spectrum. Furthermore, we show that all graphs with a perfect matching obtained from Kp,p by removing five or fewer edges are determined by their permanental spectra.
3
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The Graphs Whose Permanental Polynomials Are Symmetric

100%
Li W.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 38
|
nr 1
233-243
EN
The permanental polynomial [...] π(G,x)=∑i=0nbixn−i $\pi (G,x) = \sum\nolimits_{i = 0}^n {b_i x^{n - i} }$ of a graph G is symmetric if bi = bn−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.
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