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1
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A Note on the Permanental Roots of Bipartite Graphs

100%
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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Immanant Conversion on Symmetric Matrices

75%
Purificação Coelho M., Duffner M. A., Guterman A. E.
Special Matrices
|
2014
|
tom 2
|
nr 1
EN
Letr Σn(C) denote the space of all n χ n symmetric matrices over the complex field C. The main objective of this paper is to prove that the maps Φ : Σn(C) -> Σn (C) satisfying for any fixed irre- ducible characters X, X' -SC the condition dx(A +aB) = dχ·(Φ(Α ) + αΦ(Β)) for all matrices A,В ε Σ„(С) and all scalars a ε C are automatically linear and bijective. As a corollary of the above result we characterize all such maps Φ acting on ΣИ(С).
4
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An update on a few permanent conjectures

51%
Zhang F.
Special Matrices
|
2016
|
tom 4
|
nr 1
305-316
EN
We review and update on a few conjectures concerning matrix permanent that are easily stated, understood, and accessible to general math audience. They are: Soules permanent-on-top conjecture†, Lieb permanent dominance conjecture, Bapat and Sunder conjecture† on Hadamard product and diagonal entries, Chollet conjecture on Hadamard product, Marcus conjecture on permanent of permanents, and several other conjectures. Some of these conjectures are recently settled; some are still open.We also raise a few new questions for future study. (†conjectures have been recently settled negatively.)
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