symmetric spectrum
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Author(s):  
Willem H. Haemers ◽  
Leila Parsaei Majd

AbstractA conference matrix of order n is an $$n\times n$$ n × n matrix C with diagonal entries 0 and off-diagonal entries $$\pm 1$$ ± 1 satisfying $$CC^\top =(n-1)I$$ C C ⊤ = ( n - 1 ) I . If C is symmetric, then C has a symmetric spectrum $$\Sigma $$ Σ (that is, $$\Sigma =-\Sigma $$ Σ = - Σ ) and eigenvalues $$\pm \sqrt{n-1}$$ ± n - 1 . We show that many principal submatrices of C also have symmetric spectrum, which leads to examples of Seidel matrices of graphs (or, equivalently, adjacency matrices of complete signed graphs) with a symmetric spectrum. In addition, we show that some Seidel matrices with symmetric spectrum can be characterized by this construction.


10.37236/6241 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Martin Doležal ◽  
Jan Hladký

Hladký, Hu, and Piguet [Tilings in graphons, preprint] introduced the notions of matching and fractional vertex covers in graphons. These are counterparts to the corresponding notions in finite graphs.  Combinatorial optimization studies the structure of the matching polytope and the fractional vertex cover polytope of a graph. Here, in analogy, we initiate the study of the structure of the set of all matchings and of all fractional vertex covers in a graphon. We call these sets the matching polyton and the fractional vertex cover polyton. We also study properties of matching polytons and fractional vertex cover polytons along convergent sequences of graphons.  As an auxiliary tool of independent interest, we prove that a graphon is $r$-partite if and only if it contains no graph of chromatic number $r+1$. This in turn gives a characterization of bipartite graphons as those having a symmetric spectrum.


2018 ◽  
Vol 84 (6) ◽  
Author(s):  
Peter J. Catto

The drift surfaces of minority heated ions differ from flux surfaces due to finite poloidal gyroradius effects. As the minority poloidal gyroradius approaches radial scale lengths in the plasma, the difference between drift and flux surfaces can modify the heating and lead to a symmetric spectrum minority counter-current being driven. In response, a corresponding overall net co-current of comparable size is driven. This beneficial symmetric spectrum current drive in a tokamak is due to the parallel velocity asymmetry in the drift departure from a flux surface. As this new source of driven current is a side effect of minority heating it comes without any additional economic cost to reactor power balance. The symmetric spectrum current driven for near Maxwellian minorities is evaluated by an adjoint method and found to be modest. However, minority heating typically results in strong non-Maxwellian features on minority distributions so it may be possible to drive a significantly larger co-current. A related evaluation is performed for alpha particles in a deuterium minority heated plasma with a tritium majority. The low density of the alphas tends to keep this driven symmetric spectrum current small, but at very high heating levels a significant co-current might be driven. Other mechanisms to drive co-current with a symmetric spectrum are discussed and estimated, including asymmetric electron drag and focusing of the applied minority heating radio frequency fields.


Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5817-5826 ◽  
Author(s):  
S. Akbari ◽  
H.R. Maimani ◽  
Parsaei Majd

In this paper, we obtain the spectrum of signed complete and complete bipartite graphs whose negative edges form a matching. Moreover, we construct a family of signed complete graphs having symmetric spectrum.


2017 ◽  
Vol 24 (11) ◽  
pp. 1744-1748 ◽  
Author(s):  
Shefeng Yan ◽  
Davide Massaro ◽  
Danilo Orlando ◽  
Chengpeng Hao ◽  
Alfonso Farina

2017 ◽  
Vol 67 ◽  
pp. 131-138 ◽  
Author(s):  
Goffredo Foglia ◽  
Chengpeng Hao ◽  
Gaetano Giunta ◽  
Danilo Orlando

2017 ◽  
Vol 53 (4) ◽  
pp. 2110-2119 ◽  
Author(s):  
Goffredo Foglia ◽  
Chengpeng Hao ◽  
Alfonso Farina ◽  
Gaetano Giunta ◽  
Danilo Orlando ◽  
...  

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