542 research outputs found
Counting connected hypergraphs via the probabilistic method
In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number
of connected graphs on with edges, whenever and the nullity
tend to infinity. Asymptotic formulae for the number of connected
-uniform hypergraphs on with edges and so nullity
were proved by Karo\'nski and \L uczak for the case ,
and Behrisch, Coja-Oghlan and Kang for . Here we prove such a
formula for any fixed, and any satisfying and
as . This leaves open only the (much simpler) case
, which we will consider in future work. ( arXiv:1511.04739 )
Our approach is probabilistic. Let denote the random -uniform
hypergraph on in which each edge is present independently with
probability . Let and be the numbers of vertices and edges in
the largest component of . We prove a local limit theorem giving an
asymptotic formula for the probability that and take any given pair
of values within the `typical' range, for any in the supercritical
regime, i.e., when where
and ; our enumerative result then follows
easily.
Taking as a starting point the recent joint central limit theorem for
and , we use smoothing techniques to show that `nearby' pairs of values
arise with about the same probability, leading to the local limit theorem.
Behrisch et al used similar ideas in a very different way, that does not seem
to work in our setting.
Independently, Sato and Wormald have recently proved the special case ,
with an additional restriction on . They use complementary, more enumerative
methods, which seem to have a more limited scope, but to give additional
information when they do work.Comment: Expanded; asymptotics clarified - no significant mathematical
changes. 67 pages (including appendix
A Self-Consistent Approach to Neutral-Current Processes in Supernova Cores
The problem of neutral-current processes (neutrino scattering, pair emission,
pair absorption, axion emission, \etc) in a nuclear medium can be separated
into an expression representing the phase space of the weakly interacting
probe, and a set of dynamic structure functions of the medium. For a
non-relativistic medium we reduce the description to two structure functions
S_A(\o) and S_V(\o) of the energy transfer, representing the axial-vector
and vector interactions. is well determined by the single-nucleon
approximation while may be dominated by multiply interacting nucleons.
Unless the shape of S_A(\o) changes dramatically at high densities,
scattering processes always dominate over pair processes for neutrino transport
or the emission of right-handed states. Because the emission of right-handed
neutrinos and axions is controlled by the same medium response functions, a
consistent constraint on their properties from consideration of supernova
cooling should use the same structure functions for both neutrino transport and
exotic cooling mechanisms.Comment: 33 pages, Te
Microscopic Study of Slablike and Rodlike Nuclei: Quantum Molecular Dynamics Approach
Structure of cold dense matter at subnuclear densities is investigated by
quantum molecular dynamics (QMD) simulations. We succeeded in showing that the
phases with slab-like and rod-like nuclei etc. can be formed dynamically from
hot uniform nuclear matter without any assumptions on nuclear shape. We also
observe intermediate phases, which has complicated nuclear shapes. Geometrical
structures of matter are analyzed with Minkowski functionals, and it is found
out that intermediate phases can be characterized as ones with negative Euler
characteristic. Our result suggests the existence of these kinds of phases in
addition to the simple ``pasta'' phases in neutron star crusts.Comment: 6 pages, 4 figures, RevTex4; to be published in Phys. Rev. C Rapid
Communication (accepted version
Core Structures in Random Graphs and Hypergraphs
The k-core of a graph is its maximal subgraph with minimum degree at least k. The study of k-cores in random graphs was initiated by Bollobás in 1984 in connection to k-connected subgraphs of random graphs. Subsequently, k-cores and their properties have been extensively investigated in random graphs and hypergraphs, with the determination of the threshold for the emergence of a giant k-core, due to Pittel, Spencer and Wormald, as one of the most prominent results.
In this thesis, we obtain an asymptotic formula for the number of 2-connected graphs, as well as 2-edge-connected graphs, with given number of vertices and edges in the sparse range by exploiting properties of random 2-cores. Our results essentially cover the whole range for which asymptotic formulae were not described before. This is joint work with G. Kemkes and N. Wormald. By defining and analysing a core-type structure for uniform hypergraphs, we obtain an asymptotic formula for the number of connected 3-uniform hypergraphs with given number of vertices and edges in a sparse range. This is joint work with N. Wormald.
We also examine robustness aspects of k-cores of random graphs. More specifically, we investigate the effect that the deletion of a random edge has in the k-core as follows: we delete a random edge from the k-core, obtain the k-core of the resulting graph, and compare its order with the original k-core. For this investigation we obtain results for the giant k-core for Erdős-Rényi random graphs as well as for random graphs with minimum degree at least k and given number of vertices and edges
On retracts, absolute retracts, and folds in cographs
Let G and H be two cographs. We show that the problem to determine whether H
is a retract of G is NP-complete. We show that this problem is fixed-parameter
tractable when parameterized by the size of H. When restricted to the class of
threshold graphs or to the class of trivially perfect graphs, the problem
becomes tractable in polynomial time. The problem is also soluble when one
cograph is given as an induced subgraph of the other. We characterize absolute
retracts of cographs.Comment: 15 page
- …