Minimum Neighborhood of Alternating Group Graphs

The minimum neighborhood and combinatorial property are two important indicators of fault tolerance of a multiprocessor system. Given a graph G , θ _G (q) is the minimum number of vertices adjacent to a set of q vertices of G (1 ≤|V(G)| ). It is meant to determine θ _G (q), the minimum neighborhood problem (MNP). In this paper, we obtain θ _AGn (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 θ _AGn (2)=4n-10. Next, we prove that θ _AGn (3)=6n-16. Finally, we propose that θ _AGn (4)=8n-24.