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 r e ∗ (G ), is the minimum cardinality of reinforcement set in G that contains the edge e ∗ of the complement graph G . The average lower reinforcement number of G is defined by r av (G)=1/|E(G )| ∑e* * ∈ E (G ) r e* (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.