Unbounded Graph-Laplacians

The degree conjecture for bipartite quantum states which are normalized graph Laplacians was first put forward by Braunstein et al. [Phys. Rev. A 73 (2006) 012320]. The degree criterion, which is equivalent to PPT criterion, is simpler and more efficient to detect the separability of quantum states associated with graphs. Hassan et al. settled the degree conjecture for the separability of multipartite quantum states in [J. Math. Phys. 49 (2008) 0121105]. It is proved that the conjecture is true for pure multipartite quantum states. However, the degree condition is only necessary for separability of a class of quantum mixed states. It does not apply to all mixed states. In this paper, we show that the degree conjecture holds for the mixed quantum states of nearest point graph. As a byproduct, the degree criterion is necessary and sufficient for multipartite separability of [Formula: see text]-qubit quantum states associated with graphs.

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Sectoriality and Essential Spectrum of Non Symmetric Graph Laplacians

Complex Analysis and Operator Theory ◽

10.1007/s11785-018-0817-2 ◽

2018 ◽

Vol 13 (3) ◽

pp. 967-983 ◽

Cited By ~ 1

Author(s):

Colette Anné ◽

Marwa Balti ◽

Nabila Torki-Hamza

Keyword(s):

Essential Spectrum ◽

Symmetric Graph ◽

Graph Laplacians

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Distributed Constrained Attitude and Position Control Using Graph Laplacians

ASME 2010 Dynamic Systems and Control Conference, Volume 2 ◽

10.1115/dscc2010-4036 ◽

2010 ◽

Cited By ~ 6

Author(s):

Innocent Okoloko ◽

Yoonsoo Kim

Keyword(s):

Attitude Control ◽

Position Control ◽

Dimensional Space ◽

Stochastic Matrix ◽

Three Dimensional ◽

Laplacian Matrix ◽

Semidefinite Program ◽

Graph Theoretic ◽

Graph Laplacians ◽

Quadratically Constrained

We present a graph theoretic and optimization based method for attitude and position consensus of a team of communicating vehicles navigating in three dimensional space. Coordinated control of such vehicles has applications in planetary scale mobile sensor networks, and multiple vehicle navigation in general. Using the Laplacian matrix of the communication graph, and attitude quaternions, a synthesis of the optimal stochastic matrix that drives the attitudes to consensus, is done, by solving a constrained semidefinite program. This novel methodology attempts to extend quadratically constrained attitude control (Q-CAC), to the consensus framework. The solutions obtained are used to realize coordinated rendezvous, and formation acquisition, in the presence of static and dynamic obstacles.

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SPECTRUM OF THE LAPLACIAN OF AN ASYMMETRIC FRACTAL GRAPH

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s001309150400063x ◽

2006 ◽

Vol 49 (1) ◽

pp. 101-113 ◽

Cited By ~ 1

Author(s):

Jonathan Jordan

Keyword(s):

Similar Sequence ◽

Graph Laplacians ◽

Self Similar ◽

Spectrum Of The Laplacian

AbstractWe consider a simple self-similar sequence of graphs that does not satisfy the symmetry conditions that imply the existence of a spectral decimation property for the eigenvalues of the graph Laplacians. We show that, for this particular sequence, a very similar property to spectral decimation exists, and we obtain a complete description of the spectra of the graphs in the sequence.

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