3,013 research outputs found
Parity balance of the -th dimension edges in Hamiltonian cycles of the hypercube
Let be an integer, and let . An -th dimension
edge in the -dimensional hypercube is an edge such that
differ just at their -th entries. The parity of an -th
dimension edge \edg{v_1}{v_2} is the number of 1's modulus 2 of any of its
vertex ignoring the -th entry. We prove that the number of -th dimension
edges appearing in a given Hamiltonian cycle of with parity zero
coincides with the number of edges with parity one. As an application of this
result it is introduced and explored the conjecture of the inscribed squares in
Hamiltonian cycles of the hypercube: Any Hamiltonian cycle in contains
two opposite edges in a 4-cycle. We prove this conjecture for , and
for any Hamiltonian cycle containing more than edges in the same
dimension. This bound is finally improved considering the equi-independence
number of , which is a concept introduced in this paper for bipartite
graphs
Calculating the energy spectra of magnetic molecules: application of real- and spin-space symmetries
The determination of the energy spectra of small spin systems as for instance
given by magnetic molecules is a demanding numerical problem. In this work we
review numerical approaches to diagonalize the Heisenberg Hamiltonian that
employ symmetries; in particular we focus on the spin-rotational symmetry SU(2)
in combination with point-group symmetries. With these methods one is able to
block-diagonalize the Hamiltonian and thus to treat spin systems of
unprecedented size. In addition it provides a spectroscopic labeling by
irreducible representations that is helpful when interpreting transitions
induced by Electron Paramagnetic Resonance (EPR), Nuclear Magnetic Resonance
(NMR) or Inelastic Neutron Scattering (INS). It is our aim to provide the
reader with detailed knowledge on how to set up such a diagonalization scheme.Comment: 29 pages, many figure
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