5,817 research outputs found
Generating Functions For Kernels of Digraphs (Enumeration & Asymptotics for Nim Games)
In this article, we study directed graphs (digraphs) with a coloring
constraint due to Von Neumann and related to Nim-type games. This is equivalent
to the notion of kernels of digraphs, which appears in numerous fields of
research such as game theory, complexity theory, artificial intelligence
(default logic, argumentation in multi-agent systems), 0-1 laws in monadic
second order logic, combinatorics (perfect graphs)... Kernels of digraphs lead
to numerous difficult questions (in the sense of NP-completeness,
#P-completeness). However, we show here that it is possible to use a generating
function approach to get new informations: we use technique of symbolic and
analytic combinatorics (generating functions and their singularities) in order
to get exact and asymptotic results, e.g. for the existence of a kernel in a
circuit or in a unicircuit digraph. This is a first step toward a
generatingfunctionology treatment of kernels, while using, e.g., an approach "a
la Wright". Our method could be applied to more general "local coloring
constraints" in decomposable combinatorial structures.Comment: Presented (as a poster) to the conference Formal Power Series and
Algebraic Combinatorics (Vancouver, 2004), electronic proceeding
Rainbow eulerian multidigraphs and the product of cycles
An arc colored eulerian multidigraph with colors is rainbow eulerian if
there is an eulerian circuit in which a sequence of colors repeats. The
digraph product that refers the title was introduced by Figueroa-Centeno et al.
as follows: let be a digraph and let be a family of digraphs such
that for every . Consider any function
. Then the product is the
digraph with vertex set and if and only if and .
In this paper we use rainbow eulerian multidigraphs and permutations as a way
to characterize the -product of oriented cycles. We study the
behavior of the -product when applied to digraphs with unicyclic
components. The results obtained allow us to get edge-magic labelings of graphs
formed by the union of unicyclic components and with different magic sums.Comment: 12 pages, 5 figure
Strong coupling expansion for the Bose-Hubbard and the Jaynes-Cummings lattice model
A strong coupling expansion, based on the Kato-Bloch perturbation theory,
which has recently been proposed by Eckardt et al. [Phys. Rev. B 79, 195131]
and Teichmann et al. [Phys. Rev. B 79, 224515] is implemented in order to study
various aspects of the Bose-Hubbard and the Jaynes-Cummings lattice model. The
approach, which allows to generate numerically all diagrams up to a desired
order in the interaction strength is generalized for disordered systems and for
the Jaynes-Cummings lattice model. Results for the Bose-Hubbard and the
Jaynes-Cummings lattice model will be presented and compared with results from
VCA and DMRG. Our focus will be on the Mott insulator to superfluid transition.Comment: 29 pages, 21 figure
- …