1,293 research outputs found
The restricted -connectivity of balanced hypercubes
The restricted -connectivity of a graph , denoted by , is
defined as the minimum cardinality of a set of vertices in , if exists,
whose removal disconnects and the minimum degree of each component of
is at least . In this paper, we study the restricted -connectivity of the
balanced hypercube and determine that
for . We also obtain a sharp upper
bound of and of -dimension balanced
hypercube for (). In particular, we show that
From Graph Isoperimetric Inequality to Network Connectivity -- A New Approach
We present a new, novel approach to obtaining a network's connectivity. More
specifically, we show that there exists a relationship between a network's
graph isoperimetric properties and its conditional connectivity. A network's
connectivity is the minimum number of nodes, whose removal will cause the
network disconnected. It is a basic and important measure for the network's
reliability, hence its overall robustness. Several conditional connectivities
have been proposed in the past for the purpose of accurately reflecting various
realistic network situations, with extra connectivity being one such
conditional connectivity. In this paper, we will use isoperimetric properties
of the hypercube network to obtain its extra connectivity. The result of the
paper for the first time establishes a relationship between the age-old
isoperimetric problem and network connectivity.Comment: 17 pages, 0 figure
Structure and substructure connectivity of balanced hypercubes
The connectivity of a network directly signifies its reliability and
fault-tolerance. Structure and substructure connectivity are two novel
generalizations of the connectivity. Let be a subgraph of a connected graph
. The structure connectivity (resp. substructure connectivity) of ,
denoted by (resp. ), is defined to be the minimum
cardinality of a set of connected subgraphs in , if exists, whose
removal disconnects and each element of is isomorphic to (resp. a
subgraph of ). In this paper, we shall establish both and
of the balanced hypercube for
.Comment: arXiv admin note: text overlap with arXiv:1805.0846
The Component Connectivity of Alternating Group Graphs and Split-Stars
For an integer , the -component connectivity of a
graph , denoted by , is the minimum number of vertices
whose removal from results in a disconnected graph with at least
components or a graph with fewer than vertices. This is a natural
generalization of the classical connectivity of graphs defined in term of the
minimum vertex-cut and is a good measure of robustness for the graph
corresponding to a network. So far, the exact values of -connectivity are
known only for a few classes of networks and small 's. It has been
pointed out in~[Component connectivity of the hypercubes, Int. J. Comput. Math.
89 (2012) 137--145] that determining -connectivity is still unsolved for
most interconnection networks, such as alternating group graphs and star
graphs. In this paper, by exploring the combinatorial properties and
fault-tolerance of the alternating group graphs and a variation of the
star graphs called split-stars , we study their -component
connectivities. We obtain the following results: (i) and
for , and for
; (ii) , , and
for
Component edge connectivity of the folded hypercube
The -component edge connectivity of a non-complete graph
is the minimum number of edges whose deletion results in a graph with at
least components. In this paper, we determine the component edge
connectivity of the folded hypercube
for and , where be a
positive integer and be the decomposition of
such that and
for .Comment: The work was included in the MS thesis of the first author in [On the
component connectiviy of hypercubes and folded hypercubes, MS Thesis at
Taiyuan University of Technology, 2017
Generalized Measures of Fault Tolerance in Exchanged Hypercubes
The exchanged hypercube , proposed by Loh {\it et al.} [The
exchanged hypercube, IEEE Transactions on Parallel and Distributed Systems 16
(9) (2005) 866-874], is obtained by removing edges from a hypercube
. This paper considers a kind of generalized measures
and of fault tolerance in with and determines
for any with . The results show that at least
vertices (resp. edges) of have to be
removed to get a disconnected graph that contains no vertices of degree less
than , and generalizes some known results
The Stellar Transformation: From Interconnection Networks to Datacenter Networks
The first dual-port server-centric datacenter network, FiConn, was introduced
in 2009 and there are several others now in existence; however, the pool of
topologies to choose from remains small. We propose a new generic construction,
the stellar transformation, that dramatically increases the size of this pool
by facilitating the transformation of well-studied topologies from
interconnection networks, along with their networking properties and routing
algorithms, into viable dual-port server-centric datacenter network topologies.
We demonstrate that under our transformation, numerous interconnection networks
yield datacenter network topologies with potentially good, and easily
computable, baseline properties. We instantiate our construction so as to apply
it to generalized hypercubes and obtain the datacenter networks GQ*. Our
construction automatically yields routing algorithms for GQ* and we empirically
compare GQ* (and its routing algorithms) with the established datacenter
networks FiConn and DPillar (and their routing algorithms); this comparison is
with respect to network throughput, latency, load balancing, fault-tolerance,
and cost to build, and is with regard to all-to-all, many all-to-all,
butterfly, and random traffic patterns. We find that GQ* outperforms both
FiConn and DPillar (sometimes significantly so) and that there is substantial
scope for our stellar transformation to yield new dual-port server-centric
datacenter networks that are a considerable improvement on existing ones.Comment: Submitted to a journa
Cayley graphs and symmetric interconnection networks
These lecture notes are on automorphism groups of Cayley graphs and their
applications to optimal fault-tolerance of some interconnection networks. We
first give an introduction to automorphisms of graphs and an introduction to
Cayley graphs. We then discuss automorphism groups of Cayley graphs. We prove
that the vertex-connectivity of edge-transitive graphs is maximum possible. We
investigate the automorphism group and vertex-connectivity of some families of
Cayley graphs that have been considered for interconnection networks; we focus
on the hypercubes, folded hypercubes, Cayley graphs generated by
transpositions, and Cayley graphs from linear codes. New questions and open
problems are also discussed.Comment: A. Ganesan, "Cayley graphs and symmetric interconnection networks,"
Proceedings of the Pre-Conference Workshop on Algebraic and Applied
Combinatorics (PCWAAC 2016), 31st Annual Conference of the Ramanujan
Mathematical Society, pp. 118--170, Trichy, Tamilnadu, India, June 201
The tightly super 3-extra connectivity and 3-extra diagnosability of crossed cubes
Many multiprocessor systems have interconnection networks as underlying
topologies and an interconnection network is usually represented by a graph
where nodes represent processors and links represent communication links
between processors. In 2016, Zhang et al. proposed the -extra diagnosability
of , which restrains that every component of has at least
vertices. As an important variant of the hypercube, the -dimensional crossed
cube has many good properties. In this paper, we prove that
is tightly super 3-extra connected for and the 3-extra
diagnosability of is under the PMC model and MM
model
Connectivity and edge-bipancyclicity of hamming shell
An Any graph obtained by deleting a Hamming code of length n from a n-cube Qn
is called as a Hamming shell. It is well known that a Hamming shell is
vertex-transitive, edge-transitive, distance preserving. Moreover, it is
Hamiltonian and connected. In this paper, we prove that a Hamming shell is
edge-bipancyclic and (n-1)-connected.Comment: Total 11 pages with 5 figure
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