96,910 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
Extremal Properties of Complex Networks
We describe the structure of connected graphs with the minimum and maximum
average distance, radius, diameter, betweenness centrality, efficiency and
resistance distance, given their order and size. We find tight bounds on these
graph qualities for any arbitrary number of nodes and edges and analytically
derive the form and properties of such networks
Large Networks of Diameter Two Based on Cayley Graphs
In this contribution we present a construction of large networks of diameter
two and of order for every degree , based on Cayley
graphs with surprisingly simple underlying groups. For several small degrees we
construct Cayley graphs of diameter two and of order greater than of
Moore bound and we show that Cayley graphs of degrees
constructed in this paper are the largest
currently known vertex-transitive graphs of diameter two.Comment: 9 pages, Published in Cybernetics and Mathematics Applications in
Intelligent System
Theoretically Efficient Parallel Graph Algorithms Can Be Fast and Scalable
There has been significant recent interest in parallel graph processing due
to the need to quickly analyze the large graphs available today. Many graph
codes have been designed for distributed memory or external memory. However,
today even the largest publicly-available real-world graph (the Hyperlink Web
graph with over 3.5 billion vertices and 128 billion edges) can fit in the
memory of a single commodity multicore server. Nevertheless, most experimental
work in the literature report results on much smaller graphs, and the ones for
the Hyperlink graph use distributed or external memory. Therefore, it is
natural to ask whether we can efficiently solve a broad class of graph problems
on this graph in memory.
This paper shows that theoretically-efficient parallel graph algorithms can
scale to the largest publicly-available graphs using a single machine with a
terabyte of RAM, processing them in minutes. We give implementations of
theoretically-efficient parallel algorithms for 20 important graph problems. We
also present the optimizations and techniques that we used in our
implementations, which were crucial in enabling us to process these large
graphs quickly. We show that the running times of our implementations
outperform existing state-of-the-art implementations on the largest real-world
graphs. For many of the problems that we consider, this is the first time they
have been solved on graphs at this scale. We have made the implementations
developed in this work publicly-available as the Graph-Based Benchmark Suite
(GBBS).Comment: This is the full version of the paper appearing in the ACM Symposium
on Parallelism in Algorithms and Architectures (SPAA), 201
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