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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

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.

A Note on the Permanental Roots of Bipartite Graphs

51%

Zhang H., Liu S., Li W.

Discussiones Mathematicae Graph Theory

2014

tom 34

nr 1

49-56

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.

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

2 Li W.

1 Liu S.

1 Zhang H.

1 2018

1 2014

Wyniki wyszukiwania

The Graphs Whose Permanental Polynomials Are Symmetric

A Note on the Permanental Roots of Bipartite Graphs