Neighbor Connectivity of the Alternating Group Graph

Given a graph [Formula: see text], its neighbor connectivity is the least number of vertices whose deletion along with their neighbors results in a disconnected, complete, or empty graph. The edge neighbor connectivity is the least number of edges whose deletion along with their endpoints results in a disconnected, complete, or empty graph. In this paper, we determine the neighbor connectivity [Formula: see text] and the edge neighbor connectivity [Formula: see text] of the alternating group graph. We show that [Formula: see text], where [Formula: see text] is the [Formula: see text]-dimensional alternating group graph.

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 on the Nature Diagnosability of Alternating Group Graphs Under the PMC Model and MM* Model

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. No faulty set can contain all the neighbors of any fault-free node in the system, which is called the nature diagnosability of the system. Diagnosability of a multiprocessor system is one important study topic. As a favorable topology structure of interconnection networks, the n-dimensional alternating group graph AGn has many good properties. In this paper, we prove the following. (1) The nature diagnosability of AGn is 4n − 10 for n − 5 under the PMC model and MM* model. (2) The nature diagnosability of the 4-dimensional alternating group graph AG4 under the PMC model is 5. (3) The nature diagnosability of AG4 under the MM* model is 4.