In communications network design, network's stability is a very important concept. A network has to be constructed as possible as stable since the stability of a network shows its resistance to vulnerability. Many science and engineering problems can be represented by a network, generalization of which is a graph. Examples of problems that can be represented by a graph include: cyclic sequential circuit, organic molecule structures, mechanical structures, etc. So, a graph can be considered as a model of a communication network. Then, the notions of the graph theory can be used for the stability of a network. In the graph theory, deterministic measures of the stability are used for some parameters of graphs as connectivity, covering number, independence number and dominating number. Then, the stability of a network is defined with deterministic calculation. Today, these parameters take into consideration the neighborhood notion. Now, we consider an edge-accessibility number of a graph. Edge-accessibility is a notion which uses the neighborhood of edges (links). In this paper; we search the edge-accessibility number of a graph. We also give some theorems about the edge-accessibility using the graph operations and design an algorithm which found it with Time Complexity O(n3).