Subgraph Reliability of Alternating Group Graph With Uniform and Nonuniform Vertex Fault-Free Probabilities

As the size of a multiprocessor system grows, the probability that faults occur in this system increases. One measure of the reliability of a multiprocessor system is the probability that a fault-free subsystem of a certain size still exists with the presence of individual faults. In this paper, we use the probabilistic fault model to establish the subgraph reliability for $AG_n$, the $n$-dimensional alternating group graph. More precisely, we first analyze the probability $R_n^{n-1}(p)$ that at least one subgraph with dimension $n-1$ is fault-free in $AG_n$, when given a uniform probability of a single vertex being fault-free. Since subgraphs of $AG_n$ intersect in rather complicated manners, we resort to the principle of inclusion–exclusion by considering intersections of up to five subgraphs and obtain an upper bound of the probability. Then we consider the probabilistic fault model when the probability of a single vertex being fault-free is nonuniform, and we show that the upper bound under these two models is very close to the lower bound obtained in a previous result, and it is better than the upper bound deduced from that of the arrangement graph, which means that the upper bound we obtained is very tight.

A Note of Independent Number and Domination Number of Qn,k,m-Graph

The independent number and domination number are two essential parameters to assess the resilience of the interconnection network of multiprocessor systems which is usually modeled by a graph. The independent number, denoted by [Formula: see text], of a graph [Formula: see text] is the maximum cardinality of any subset [Formula: see text] such that no two elements in [Formula: see text] are adjacent in [Formula: see text]. The domination number, denoted by [Formula: see text], of a graph [Formula: see text] is the minimum cardinality of any subset [Formula: see text] such that every vertex in [Formula: see text] is either in [Formula: see text] or adjacent to an element of [Formula: see text]. But so far, determining the independent number and domination number of a graph is still an NPC problem. Therefore, it is of utmost importance to determine the number of independent and domination number of some special networks with potential applications in multiprocessor system. In this paper, we firstly resolve the exact values of independent number and upper and lower bound of domination number of the [Formula: see text]-graph, a common generalization of various popular interconnection networks. Besides, as by-products, we derive the independent number and domination number of [Formula: see text]-star graph [Formula: see text], [Formula: see text]-arrangement graph [Formula: see text], as well as three special graphs.

g-Good-Neighbor Diagnosability of Arrangement Graphs under the PMC Model and MM* Model

Diagnosability of a multiprocessor system is an important research topic. The system and interconnection network has a underlying topology, which usually presented by a graph G = ( V , E ) . In 2012, a measurement for fault tolerance of the graph was proposed by Peng et al. This measurement is called the g-good-neighbor diagnosability that restrains every fault-free node to contain at least g fault-free neighbors. Under the PMC model, to diagnose the system, two adjacent nodes in G are can perform tests on each other. Under the MM model, to diagnose the system, a node sends the same task to two of its neighbors, and then compares their responses. The MM* is a special case of the MM model and each node must test its any pair of adjacent nodes of the system. As a famous topology structure, the ( n , k ) -arrangement graph A n , k , has many good properties. In this paper, we give the g-good-neighbor diagnosability of A n , k under the PMC model and MM* model.

The Super Spanning Connectivity of Arrangement Graphs

A [Formula: see text]-container [Formula: see text] of a graph [Formula: see text] is a set of [Formula: see text] internally disjoint paths between [Formula: see text] and [Formula: see text]. A [Formula: see text]-container [Formula: see text] of [Formula: see text] is a [Formula: see text]-container if it is a spanning subgraph of [Formula: see text]. A graph [Formula: see text] is [Formula: see text]-connected if there exists a [Formula: see text]-container between any two different vertices of G. A [Formula: see text]-regular graph [Formula: see text] is super spanning connected if [Formula: see text] is [Formula: see text]-container for all [Formula: see text]. In this paper, we prove that the arrangement graph [Formula: see text] is super spanning connected if [Formula: see text] and [Formula: see text].

Link Failure Tolerance in the Arrangement Graphs

The arrangement graph An,k is one of the attractive underlying topologies for distributed systems. Let fm(n, k) be the minimum number of faulty links that make every sub-arrangement graph An-m,k-m faulty in An,k under link failure model. In this paper, we proved that [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text].

The number of shortest paths in the (n, k)-star graph

We count the shortest paths, not necessarily disjoint, between any two vertices in an (n, k)-star graph by counting the minimum factorizations of a permutation in terms of the transpositions corresponding to edges in that graph. This result generalizes a previous one for the star graph, and can be applied to obtain the number of the shortest paths between a pair of vertices in some of the other structures closely related to the (n, k)-star graph, such as the alternating group networks. Furthermore, the techniques made use of in this paper can be applied to solve the same problem for some of the other structures such as the arrangement graph.