5,198 research outputs found
Relationship between Conditional Diagnosability and 2-extra Connectivity of Symmetric Graphs
The conditional diagnosability and the 2-extra connectivity are two important
parameters to measure ability of diagnosing faulty processors and
fault-tolerance in a multiprocessor system. The conditional diagnosability
of is the maximum number for which is conditionally
-diagnosable under the comparison model, while the 2-extra connectivity
of a graph is the minimum number for which there is a
vertex-cut with such that every component of has at least
vertices. A quite natural problem is what is the relationship between the
maximum and the minimum problem? This paper partially answer this problem by
proving for a regular graph with some acceptable
conditions. As applications, the conditional diagnosability and the 2-extra
connectivity are determined for some well-known classes of vertex-transitive
graphs, including, star graphs, -star graphs, alternating group
networks, -arrangement graphs, alternating group graphs, Cayley graphs
obtained from transposition generating trees, bubble-sort graphs, -ary
-cube networks and dual-cubes. Furthermore, many known results about these
networks are obtained directly
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
Super edge-connectivity and matching preclusion of data center networks
Edge-connectivity is a classic measure for reliability of a network in the
presence of edge failures. -restricted edge-connectivity is one of the
refined indicators for fault tolerance of large networks. Matching preclusion
and conditional matching preclusion are two important measures for the
robustness of networks in edge fault scenario. In this paper, we show that the
DCell network is super- for and ,
super- for and , or and , and
super- for and . Moreover, as an application of
-restricted edge-connectivity, we study the matching preclusion number and
conditional matching preclusion number, and characterize the corresponding
optimal solutions of . In particular, we have shown that is
isomorphic to the -star graph for .Comment: 20 pages, 1 figur
On the -good-neighbor connectivity of graphs
Connectivity and diagnosability are two important parameters for the fault
tolerant of an interconnection network . In 1996, F\`{a}brega and Fiol
proposed the -good-neighbor connectivity of . In this paper, we show that
for , and graphs with and
trees with for are
characterized, respectively. In the end, we get the three extremal results for
the -good-neighbor connectivity.Comment: 14 pages, 2 figures. arXiv admin note: substantial text overlap with
arXiv:1904.06527; text overlap with arXiv:1609.08885, arXiv:1612.05381 by
other author
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
Fault Diagnosability of Arrangement Graphs
The growing size of the multiprocessor system increases its vulnerability to
component failures. It is crucial to locate and to replace the faulty
processors to maintain a system's high reliability. The fault diagnosis is the
process of identifying faulty processors in a system through testing. This
paper shows that the largest connected component of the survival graph contains
almost all remaining vertices in the -arrangement graph when
the number of moved faulty vertices is up to twice or three times the
traditional connectivity. Based on this fault resiliency, we establishes that
the conditional diagnosability of under the comparison model. We
prove that for , , the conditional diagnosability of
is ; the conditional diagnosability of is
for .Comment: 21 pages, 2 figures, 37 refrence
Fractional matching preclusion for restricted hypercube-like graphs
The restricted hypercube-like graphs, variants of the hypercube, were
proposed as desired interconnection networks of parallel systems. The matching
preclusion number of a graph is the minimum number of edges whose deletion
results in the graph with neither perfect matchings nor almost perfect
matchings. The fractional perfect matching preclusion and fractional strong
perfect matching preclusion are generalizations of the concept matching
preclusion. In this paper, we obtain fractional matching preclusion number and
fractional strong matching preclusion numbers of restricted hypercube-like
graphs, which extend some known results
NP-completeness of anti-Kekul\'e and matching preclusion problems
Anti-Kekul\'{e} problem is a concept of chemical graph theory to measure
stability of the molecule by precluding its Kekul\'{e} structure. Matching
preclusion and conditional matching preclusion problems were proposed as
measures of robustness in the event of edge failure in interconnection
networks. It is known that matching preclusion problem on bipartite graphs is
NP-complete. In this paper, we first show that conditional matching preclusion
problem and anti-Kekul\'{e} problem are NP-complete, respectively. We then
generalize matching preclusion problem to -restricted matching preclusion
problem and prove its NP-completeness. Moreover, we give some sufficient
conditions to compute -restricted matching preclusion numbers of regular
graphs. As applications, -restricted matching preclusion numbers of complete
graphs, hypercubes and hyper Petersen networks are determined.Comment: 16 pages, 1 figur
The 4-Component Connectivity of Alternating Group Networks
The -component connectivity (or -connectivity for short) 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 generalization is a
natural extension of the classical connectivity defined in term of minimum
vertex-cut. As an application, the -connectivity can be used to assess
the vulnerability of a graph corresponding to the underlying topology of an
interconnection network, and thus is an important issue for reliability and
fault tolerance of the network. So far, only a little knowledge of results have
been known on -connectivity for particular classes of graphs and small
's. In a previous work, we studied the -connectivity on
-dimensional alternating group networks and obtained the result
for . In this sequel, we continue the work
and show that for
On the eigenvalues of certain Cayley graphs and arrangement graphs
In this paper, we show that the eigenvalues of certain classes of Cayley
graphs are integers. The (n,k,r)-arrangement graph A(n,k,r) is a graph with all
the k-permutations of an n-element set as vertices where two k-permutations are
adjacent if they differ in exactly r positions. We establish a relation between
the eigenvalues of the arrangement graphs and the eigenvalues of certain Cayley
graphs. As a result, the conjecture on integrality of eigenvalues of A(n,k,1)
follows.Comment: 12 pages, final versio
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