Extreme graphs with given order and edge-neighbor-scattering number

Let G= (V , E) be a graph. We call X(⊆ E(G)) an edge subversion strategy of G if the closed neighborhood of X is deleted from G . The survival subgraph is denoted byG / X . An edge subversion strategy X is called an edge-cut-strategy of G if G / X is disconnected, or is a single vertex, or is empty. The edge-neighbor-scattering number (ENS) of G is defined to be ENS(G)= max{ω(G/X ) - |X|}, where X is an edge-cut-strategy of G , ω(G / X ) is the number of the connected components of G / X . This parameter can be used to measure the neighbor invulnerability of some special networks such as spy networks, spread of high risk virus networks, etc. In this paper we study the extreme graph problems on ENS: determine the maximum and the minimum edge numbers of graphs with given order and ENS.