187 research outputs found
The degree-diameter problem for sparse graph classes
The degree-diameter problem asks for the maximum number of vertices in a
graph with maximum degree and diameter . For fixed , the answer
is . We consider the degree-diameter problem for particular
classes of sparse graphs, and establish the following results. For graphs of
bounded average degree the answer is , and for graphs of
bounded arboricity the answer is \Theta(\Delta^{\floor{k/2}}), in both cases
for fixed . For graphs of given treewidth, we determine the the maximum
number of vertices up to a constant factor. More precise bounds are given for
graphs of given treewidth, graphs embeddable on a given surface, and
apex-minor-free graphs
Pseudograph associahedra
Given a simple graph G, the graph associahedron KG is a simple polytope whose
face poset is based on the connected subgraphs of G. This paper defines and
constructs graph associahedra in a general context, for pseudographs with loops
and multiple edges, which are also allowed to be disconnected. We then consider
deformations of pseudograph associahedra as their underlying graphs are altered
by edge contractions and edge deletions.Comment: 25 pages, 22 figure
Systematic Topology Analysis and Generation Using Degree Correlations
We present a new, systematic approach for analyzing network topologies. We
first introduce the dK-series of probability distributions specifying all
degree correlations within d-sized subgraphs of a given graph G. Increasing
values of d capture progressively more properties of G at the cost of more
complex representation of the probability distribution. Using this series, we
can quantitatively measure the distance between two graphs and construct random
graphs that accurately reproduce virtually all metrics proposed in the
literature. The nature of the dK-series implies that it will also capture any
future metrics that may be proposed. Using our approach, we construct graphs
for d=0,1,2,3 and demonstrate that these graphs reproduce, with increasing
accuracy, important properties of measured and modeled Internet topologies. We
find that the d=2 case is sufficient for most practical purposes, while d=3
essentially reconstructs the Internet AS- and router-level topologies exactly.
We hope that a systematic method to analyze and synthesize topologies offers a
significant improvement to the set of tools available to network topology and
protocol researchers.Comment: Final versio
New upper bounds for the number of embeddings of minimally rigid graphs
By definition, a rigid graph in (or on a sphere) has a finite
number of embeddings up to rigid motions for a given set of edge length
constraints. These embeddings are related to the real solutions of an algebraic
system. Naturally, the complex solutions of such systems extend the notion of
rigidity to . A major open problem has been to obtain tight upper
bounds on the number of embeddings in , for a given number
of vertices, which obviously also bound their number in .
Moreover, in most known cases, the maximal numbers of embeddings in
and coincide. For decades, only the trivial bound
of was known on the number of embeddings.Recently, matrix
permanent bounds have led to a small improvement for . This work
improves upon the existing upper bounds for the number of embeddings in
and , by exploiting outdegree-constrained orientations on a
graphical construction, where the proof iteratively eliminates vertices or
vertex paths. For the most important cases of and , the new bounds
are and , respectively. In general, the
recent asymptotic bound mentioned above is improved by a factor of . Besides being the first substantial improvement upon a long-standing
upper bound, our method is essentially the first general approach relying on
combinatorial arguments rather than algebraic root counts
The dynamics of pseudographs in convex Hamiltonian systems
We study the evolution, under convex Hamiltonian flows on cotangent bundles
of compact manifolds, of certain distinguished subsets of the phase space.
These subsets are generalizations of Lagrangian graphs, we call them
pseudographs. They emerge in a natural way from Fathi's weak KAM theory. By
this method, we find various orbits which connect prescribed regions of the
phase space. Our study is inspired by works of John Mather. As an application,
we obtain the existence of diffusion in a large class of a priori unstable
systems and provide a solution to the large gap problem. We hope that our
method will have applications to more examples
Drawing a Graph in a Hypercube
A -dimensional hypercube drawing of a graph represents the vertices by
distinct points in , such that the line-segments representing the
edges do not cross. We study lower and upper bounds on the minimum number of
dimensions in hypercube drawing of a given graph. This parameter turns out to
be related to Sidon sets and antimagic injections.Comment: Submitte
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