Minimum Neighborhood of Alternating Group Graphs

Abstract

The minimum neighborhood and combinatorial property are two important indicators of fault tolerance of a multiprocessor system. Given a graph $G$ , $\theta _{G}(q)$ is the minimum number of vertices adjacent to a set of $q$ vertices of $G$ ( $1\leq q\leq |V(G)|$ ). It is meant to determine $\theta _{G}(q)$ , the minimum neighborhood problem (MNP). In this paper, we obtain $\theta _{AG_{n}}(q)$ for an independent set with size $q$ in an $n$ -dimensional alternating group graph $AG_{n}$ , a well-known interconnection network for multiprocessor systems. We first propose some combinatorial properties of $AG_{n}$ . Then, we study the MNP for an independent set of two vertices and obtain that $\theta _{AG_{n}}(2)=4n-10$ . Next, we prove that $\theta _{AG_{n}}(3)=6n-16$ . Finally, we propose that $\theta _{AG_{n}}(4)=8n-24$ .