4,707 research outputs found
Further results on random cubic planar graphs
We provide precise asymptotic estimates for the number of several classes of labeled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by Bodirsky and coworkers. We revisit their work and obtain new results on the enumeration of cubic planar graphs and on random cubic planar graphs. In particular, we determine the exact probability of a random cubic planar graph being connected, and we show that the distribution of the number of triangles in random cubic planar graphs is asymptotically normal with linear expectation and variance. To the best of our knowledge, this is the first time one is able to determine the asymptotic distribution for the number of copies of a fixed graph containing a cycle in classes of random planar graphs arising from planar maps.Peer ReviewedPostprint (author's final draft
The First Thirty Years of Large-N Gauge Theory
I review some developments in the large-N gauge theory since 1974. The main
attention is payed to: multicolor QCD, matrix models, loop equations, reduced
models, 2D quantum gravity, free random variables, noncommutative theories,
AdS/CFT correspondence.Comment: 13.1pp., Latex, 2 figs; v2: 2 refs added. Talk at Large Nc QCD 200
Statistics of planar graphs viewed from a vertex: A study via labeled trees
We study the statistics of edges and vertices in the vicinity of a reference
vertex (origin) within random planar quadrangulations and Eulerian
triangulations. Exact generating functions are obtained for theses graphs with
fixed numbers of edges and vertices at given geodesic distances from the
origin. Our analysis relies on bijections with labeled trees, in which the
labels encode the information on the geodesic distance from the origin. In the
case of infinitely large graphs, we give in particular explicit formulas for
the probabilities that the origin have given numbers of neighboring edges
and/or vertices, as well as explicit values for the corresponding moments.Comment: 36 pages, 15 figures, tex, harvmac, eps
On the Universality of Matrix Models for Random Surfaces
We present an alternative procedure to eliminate irregular contributions in
the perturbation expansion of c=0-matrix models representing the sum over
triangulations of random surfaces, thereby reproducing the results of Tutte [1]
and Brezin et al. [2] for the planar model. The advantage of this method is
that the universality of the critical exponents can be proven from general
features of the model alone without explicit determination of the free energy
and therefore allows for several straightforward generalizations including
cases with non-vanishing central charge c< 1.Comment: 9 pages, 3 figure
Drawings of Planar Graphs with Few Slopes and Segments
We study straight-line drawings of planar graphs with few segments and few
slopes. Optimal results are obtained for all trees. Tight bounds are obtained
for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every
3-connected plane graph on vertices has a plane drawing with at most
segments and at most slopes. We prove that every cubic
3-connected plane graph has a plane drawing with three slopes (and three bends
on the outerface). In a companion paper, drawings of non-planar graphs with few
slopes are also considered.Comment: This paper is submitted to a journal. A preliminary version appeared
as "Really Straight Graph Drawings" in the Graph Drawing 2004 conference. See
http://arxiv.org/math/0606446 for a companion pape
Single-Strip Triangulation of Manifolds with Arbitrary Topology
Triangle strips have been widely used for efficient rendering. It is
NP-complete to test whether a given triangulated model can be represented as a
single triangle strip, so many heuristics have been proposed to partition
models into few long strips. In this paper, we present a new algorithm for
creating a single triangle loop or strip from a triangulated model. Our method
applies a dual graph matching algorithm to partition the mesh into cycles, and
then merges pairs of cycles by splitting adjacent triangles when necessary. New
vertices are introduced at midpoints of edges and the new triangles thus formed
are coplanar with their parent triangles, hence the visual fidelity of the
geometry is not changed. We prove that the increase in the number of triangles
due to this splitting is 50% in the worst case, however for all models we
tested the increase was less than 2%. We also prove tight bounds on the number
of triangles needed for a single-strip representation of a model with holes on
its boundary. Our strips can be used not only for efficient rendering, but also
for other applications including the generation of space filling curves on a
manifold of any arbitrary topology.Comment: 12 pages, 10 figures. To appear at Eurographics 200
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