2 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
The generalized connectivity of some regular graphs
The generalized -connectivity of a graph is a
parameter that can measure the reliability of a network to connect any
vertices in , which is proved to be NP-complete for a general graph . Let
and denote the maximum number of
edge-disjoint trees in such that
for any and . For an integer with , the {\em generalized
-connectivity} of a graph is defined as and .
In this paper, we study the generalized -connectivity of some general
-regular and -connected graphs constructed recursively and obtain
that , which attains the upper bound of
[Discrete Mathematics 310 (2010) 2147-2163] given by Li {\em et al.} for
. As applications of the main result, the generalized -connectivity
of many famous networks such as the alternating group graph , the
-ary -cube , the split-star network and the
bubble-sort-star graph etc. can be obtained directly.Comment: 19 pages, 6 figure