5,439 research outputs found
Generalized constructive tree weights
The Loop Vertex Expansion (LVE) is a quantum field theory (QFT) method which
explicitly computes the Borel sum of Feynman perturbation series. This LVE
relies in a crucial way on symmetric tree weights which define a measure on the
set of spanning trees of any connected graph. In this paper we generalize this
method by defining new tree weights. They depend on the choice of a partition
of a set of vertices of the graph, and when the partition is non-trivial, they
are no longer symmetric under permutation of vertices. Nevertheless we prove
they have the required positivity property to lead to a convergent LVE; in
fact, we formulate this positivity property precisely for the first time. Our
generalized tree weights are inspired by the Brydges-Battle-Federbush work on
cluster expansions and could be particularly suited to the computation of
connected functions in QFT. Several concrete examples are explicitly given.Comment: 22 pages, 2 figure
Recognizing Partial Cubes in Quadratic Time
We show how to test whether a graph with n vertices and m edges is a partial
cube, and if so how to find a distance-preserving embedding of the graph into a
hypercube, in the near-optimal time bound O(n^2), improving previous O(nm)-time
solutions.Comment: 25 pages, five figures. This version significantly expands previous
versions, including a new report on an implementation of the algorithm and
experiments with i
Grassmann Integral Representation for Spanning Hyperforests
Given a hypergraph G, we introduce a Grassmann algebra over the vertex set,
and show that a class of Grassmann integrals permits an expansion in terms of
spanning hyperforests. Special cases provide the generating functions for
rooted and unrooted spanning (hyper)forests and spanning (hyper)trees. All
these results are generalizations of Kirchhoff's matrix-tree theorem.
Furthermore, we show that the class of integrals describing unrooted spanning
(hyper)forests is induced by a theory with an underlying OSP(1|2)
supersymmetry.Comment: 50 pages, it uses some latex macros. Accepted for publication on J.
Phys.
Causal and homogeneous networks
Growing networks have a causal structure. We show that the causality strongly
influences the scaling and geometrical properties of the network. In particular
the average distance between nodes is smaller for causal networks than for
corresponding homogeneous networks. We explain the origin of this effect and
illustrate it using as an example a solvable model of random trees. We also
discuss the issue of stability of the scale-free node degree distribution. We
show that a surplus of links may lead to the emergence of a singular node with
the degree proportional to the total number of links. This effect is closely
related to the backgammon condensation known from the balls-in-boxes model.Comment: short review submitted to AIP proceedings, CNET2004 conference;
changes in the discussion of the distance distribution for growing trees,
Fig. 6-right change
- …