4,364 research outputs found
Constructions of Large Graphs on Surfaces
We consider the degree/diameter problem for graphs embedded in a surface,
namely, given a surface and integers and , determine the
maximum order of a graph embeddable in with
maximum degree and diameter . We introduce a number of
constructions which produce many new largest known planar and toroidal graphs.
We record all these graphs in the available tables of largest known graphs.
Given a surface of Euler genus and an odd diameter , the
current best asymptotic lower bound for is given by
Our constructions produce
new graphs of order \begin{cases}6\Delta^{\lfloor k/2\rfloor}& \text{if
$\Sigma$ is the Klein bottle}\\
\(\frac{7}{2}+\sqrt{6g+\frac{1}{4}}\)\Delta^{\lfloor k/2\rfloor}&
\text{otherwise,}\end{cases} thus improving the former value by a factor of
4.Comment: 15 pages, 7 figure
Exploring complex networks via topological embedding on surfaces
We demonstrate that graphs embedded on surfaces are a powerful and practical
tool to generate, characterize and simulate networks with a broad range of
properties. Remarkably, the study of topologically embedded graphs is
non-restrictive because any network can be embedded on a surface with
sufficiently high genus. The local properties of the network are affected by
the surface genus which, for example, produces significant changes in the
degree distribution and in the clustering coefficient. The global properties of
the graph are also strongly affected by the surface genus which is constraining
the degree of interwoveness, changing the scaling properties from
large-world-kind (small genus) to small- and ultra-small-world-kind (large
genus). Two elementary moves allow the exploration of all networks embeddable
on a given surface and naturally introduce a tool to develop a statistical
mechanics description. Within such a framework, we study the properties of
topologically-embedded graphs at high and low `temperatures' observing the
formation of increasingly regular structures by cooling the system. We show
that the cooling dynamics is strongly affected by the surface genus with the
manifestation of a glassy-like freezing transitions occurring when the amount
of topological disorder is low.Comment: 18 pages, 7 figure
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
The degree/diameter problem in maximal planar bipartite graphs
The (Δ,D)(Δ,D) (degree/diameter) problem consists of finding the largest possible number of vertices nn among all the graphs with maximum degree ΔΔ and diameter DD. We consider the (Δ,D)(Δ,D) problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the (Δ,2)(Δ,2) problem, the number of vertices is n=Δ+2n=Δ+2; and for the (Δ,3)(Δ,3) problem, n=3Δ−1n=3Δ−1 if ΔΔ is odd and n=3Δ−2n=3Δ−2 if ΔΔ is even. Then, we prove that, for the general case of the (Δ,D)(Δ,D) problem, an upper bound on nn is approximately 3(2D+1)(Δ−2)⌊D/2⌋3(2D+1)(Δ−2)⌊D/2⌋, and another one is C(Δ−2)⌊D/2⌋C(Δ−2)⌊D/2⌋ if Δ≥DΔ≥D and CC is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on nn for maximal planar bipartite graphs, which is approximately (Δ−2)k(Δ−2)k if D=2kD=2k, and 3(Δ−3)k3(Δ−3)k if D=2k+1D=2k+1, for ΔΔ and DD sufficiently large in both cases.Peer ReviewedPostprint (published version
On the maximum order of graphs embedded in surfaces
The maximum number of vertices in a graph of maximum degree and
fixed diameter is upper bounded by . If we
restrict our graphs to certain classes, better upper bounds are known. For
instance, for the class of trees there is an upper bound of
for a fixed . The main result of
this paper is that graphs embedded in surfaces of bounded Euler genus
behave like trees, in the sense that, for large , such graphs have
orders bounded from above by begin{cases} c(g+1)(\Delta-1)^{\lfloor
k/2\rfloor} & \text{if $k$ is even} c(g^{3/2}+1)(\Delta-1)^{\lfloor k/2\rfloor}
& \text{if $k$ is odd}, \{cases} where is an absolute constant. This
result represents a qualitative improvement over all previous results, even for
planar graphs of odd diameter . With respect to lower bounds, we construct
graphs of Euler genus , odd diameter , and order
for some absolute constant
. Our results answer in the negative a question of Miller and
\v{S}ir\'a\v{n} (2005).Comment: 13 pages, 3 figure
Self-similar planar graphs as models for complex networks
In this paper we introduce a family of planar, modular and self-similar
graphs which have small-world and scale-free properties. The main parameters of
this family are comparable to those of networks associated to complex systems,
and therefore the graphs are of interest as mathematical models for these
systems. As the clustering coefficient of the graphs is zero, this family is an
explicit construction that does not match the usual characterization of
hierarchical modular networks, namely that vertices have clustering values
inversely proportional to their degrees.Comment: 10 pages, submitted to 19th International Workshop on Combinatorial
Algorithms (IWOCA 2008
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