3,319 research outputs found
Asymptotic enumeration of 2-covers and line graphs
In this paper we find asymptotic enumerations for the number of line graphs
on -labelled vertices and for different types of related combinatorial
objects called 2-covers.
We find that the number of 2-covers, , and proper 2-covers, , on
both have asymptotic growth where is the th Bell number, while the number of
restricted 2-covers, , restricted, proper 2-covers on , , and
line graphs , all have growth
In our proofs we use probabilistic arguments for the unrestricted types of
2-covers and and generating function methods for the restricted types of
2-covers and line graphs
Automatic enumeration of regular objects
We describe a framework for systematic enumeration of families combinatorial
structures which possess a certain regularity. More precisely, we describe how
to obtain the differential equations satisfied by their generating series.
These differential equations are then used to determine the initial counting
sequence and for asymptotic analysis. The key tool is the scalar product for
symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer
Sequence
Vertex cover problem studied by cavity method: Analytics and population dynamics
We study the vertex cover problem on finite connectivity random graphs by
zero-temperature cavity method. The minimum vertex cover corresponds to the
ground state(s) of a proposed Ising spin model. When the connectivity
c>e=2.718282, there is no state for this system as the reweighting parameter y,
which takes a similar role as the inverse temperature \beta in conventional
statistical physics, approaches infinity; consequently the ground state energy
is obtained at a finite value of y when the free energy function attains its
maximum value. The minimum vertex cover size at given c is estimated using
population dynamics and compared with known rigorous bounds and numerical
results. The backbone size is also calculated.Comment: 7 pages (including 3 figures and 1 table), REVTeX4 forma
The enumeration of planar graphs via Wick's theorem
A seminal technique of theoretical physics called Wick's theorem interprets
the Gaussian matrix integral of the products of the trace of powers of
Hermitian matrices as the number of labelled maps with a given degree sequence,
sorted by their Euler characteristics. This leads to the map enumeration
results analogous to those obtained by combinatorial methods. In this paper we
show that the enumeration of the graphs embeddable on a given 2-dimensional
surface (a main research topic of contemporary enumerative combinatorics) can
also be formulated as the Gaussian matrix integral of an ice-type partition
function. Some of the most puzzling conjectures of discrete mathematics are
related to the notion of the cycle double cover. We express the number of the
graphs with a fixed directed cycle double cover as the Gaussian matrix integral
of an Ihara-Selberg-type function.Comment: 23 pages, 2 figure
Strings from Feynman Graph counting : without large N
A well-known connection between n strings winding around a circle and
permutations of n objects plays a fundamental role in the string theory of
large N two dimensional Yang Mills theory and elsewhere in topological and
physical string theories. Basic questions in the enumeration of Feynman graphs
can be expressed elegantly in terms of permutation groups. We show that these
permutation techniques for Feynman graph enumeration, along with the Burnside
counting lemma, lead to equalities between counting problems of Feynman graphs
in scalar field theories and Quantum Electrodynamics with the counting of
amplitudes in a string theory with torus or cylinder target space. This string
theory arises in the large N expansion of two dimensional Yang Mills and is
closely related to lattice gauge theory with S_n gauge group. We collect and
extend results on generating functions for Feynman graph counting, which
connect directly with the string picture. We propose that the connection
between string combinatorics and permutations has implications for QFT-string
dualities, beyond the framework of large N gauge theory.Comment: 55 pages + 10 pages Appendices, 23 figures ; version 2 - typos
correcte
- β¦