The average lower reinforcement number of a graph

Let G = (V(G),E(G)) be a simple undirected graph. The reinforcement number of a graph is a vulnerability parameter of a graph. We have investigated a refinement that involves the average lower reinforcement number of this parameter. The lower reinforcement number, denoted by re∗(G), is the minimum cardinality of reinforcement set in G that contains the edge e∗ of the complement graph G̅ . The average lower reinforcement number of G is defined by rav(G)=1/|E(G̅)| ∑e** ∈ E(G̅) re*(G) .In this paper, we define the average lower reinforcement number of a graph and we present the exact values for some well−known graph families.