Average covering number for some graphs

The interconnection networks are modeled by means of graphs to determine the reliability and vulnerability. There are lots of parameters that are used to determine vulnerability. The average covering number is one of them which is denoted by $ \overline{\beta }(G)$, where G is simple, connected and undirected graph of order n ≥ 2. In a graph G = (V(G), E(G)) a subset $ {S}_v\subseteq V(G)$ of vertices is called a cover set of G with respect to v or a local covering set of vertex v, if each edge of the graph is incident to at least one vertex of Sv. The local covering number with respect to v is the minimum cardinality of among the Sv sets and denoted by βv. The average covering number of a graph G is defined as β̅(G) = 1/|v(G)| ∑ν∈v(G)βν In this paper, the average covering numbers of kth power of a cycle $ {C}_n^k$ and Pn □ Pm, Pn □ Cm, cartesian product of Pn and Pm, cartesian product of Pn and Cm are given, respectively.