3,873 research outputs found
Geometry of discrete and continuous bounded surfaces
We work on reconstructing discrete and continuous surfaces with boundaries using length constraints. First, for a bounded discrete surface, we discuss the rigidity and number of embeddings in three-dimensional space, modulo rigid transformations, for given real edge lengths. Our work mainly considers the maximal number of embeddings of rigid graphs in three-dimensional space for specific geometries (annulus, strip). We modify a commonly used semi-algebraic, geometrical formulation using Bézout\u27s theorem, from Euclidean distances corresponding to edge lengths. We suggest a simple way to construct a rigid graph having a finite upper bound. We also implement a generalization of counting embeddings for graphs by segmenting multiple rigid graphs in d-dimensional space. Our computational methodology uses vector and matrix operations and can work best with a relatively small number of points
Surface embedding, topology and dualization for spin networks
Spin networks are graphs derived from 3nj symbols of angular momentum. The
surface embedding, the topology and dualization of these networks are
considered. Embeddings into compact surfaces include the orientable sphere S^2
and the torus T, and the not orientable projective space P^2 and Klein's bottle
K. Two families of 3nj graphs admit embeddings of minimal genus into S^2 and
P^2. Their dual 2-skeletons are shown to be triangulations of these surfaces.Comment: LaTeX 17 pages, 6 eps figures (late submission to arxiv.org
On the maximal number of real embeddings of minimally rigid graphs in , and
Rigidity theory studies the properties of graphs that can have rigid
embeddings in a euclidean space or on a sphere and which in
addition satisfy certain edge length constraints. One of the major open
problems in this field is to determine lower and upper bounds on the number of
realizations with respect to a given number of vertices. This problem is
closely related to the classification of rigid graphs according to their
maximal number of real embeddings.
In this paper, we are interested in finding edge lengths that can maximize
the number of real embeddings of minimally rigid graphs in the plane, space,
and on the sphere. We use algebraic formulations to provide upper bounds. To
find values of the parameters that lead to graphs with a large number of real
realizations, possibly attaining the (algebraic) upper bounds, we use some
standard heuristics and we also develop a new method inspired by coupler
curves. We apply this new method to obtain embeddings in . One of
its main novelties is that it allows us to sample efficiently from a larger
number of parameters by selecting only a subset of them at each iteration.
Our results include a full classification of the 7-vertex graphs according to
their maximal numbers of real embeddings in the cases of the embeddings in
and , while in the case of we achieve this
classification for all 6-vertex graphs. Additionally, by increasing the number
of embeddings of selected graphs, we improve the previously known asymptotic
lower bound on the maximum number of realizations. The methods and the results
concerning the spatial embeddings are part of the proceedings of ISSAC 2018
(Bartzos et al, 2018)
Monotone Maps, Sphericity and Bounded Second Eigenvalue
We consider {\em monotone} embeddings of a finite metric space into low
dimensional normed space. That is, embeddings that respect the order among the
distances in the original space. Our main interest is in embeddings into
Euclidean spaces. We observe that any metric on points can be embedded into
, while, (in a sense to be made precise later), for almost every
-point metric space, every monotone map must be into a space of dimension
.
It becomes natural, then, to seek explicit constructions of metric spaces
that cannot be monotonically embedded into spaces of sublinear dimension. To
this end, we employ known results on {\em sphericity} of graphs, which suggest
one example of such a metric space - that defined by a complete bipartitegraph.
We prove that an -regular graph of order , with bounded diameter
has sphericity , where is the second
largest eigenvalue of the adjacency matrix of the graph, and 0 < \delta \leq
\half is constant. We also show that while random graphs have linear
sphericity, there are {\em quasi-random} graphs of logarithmic sphericity.
For the above bound to be linear, must be constant. We show that
if the second eigenvalue of an -regular graph is bounded by a constant,
then the graph is close to being complete bipartite. Namely, its adjacency
matrix differs from that of a complete bipartite graph in only
entries. Furthermore, for any 0 < \delta < \half, and , there are
only finitely many -regular graphs with second eigenvalue at most
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