Some vulnerability measures in graphs and their relationships

In the network (communication, computer, electricity, transportation etc.), after the failure of certain centers or connecting lines, the network shows the resistance to disruption of a operation, which is measured with the vulnerability in graph theory. To determine the value of vulnerability of a network, a graph is modelled with a network whose centers are corresponded to the vertices of a graph and whose links are corresponded to the edges of a graph. The edge neighbor rupture degree of a connected graph G is defined as ENR(G)=max┬(S⊆E(G) )⁡〖{w(G/S)-|S|-m(G/S) ∶ w(G/S)≥1}〗 where S is any edge subversion strategy of G, w(G/S) is the number of components of G/S and m(G/S) is the maximum order of the components of G/S. On the other hand, the edge scattering number of a connected graph G is defined as es(G)=max┬(S⊆E(G) )⁡{w(G-S)-|S| ∶ w(G-S)>1} where S is any edge-cut set of G, w(G-S) is the number of the components of G-S. In this thesis, the edge neighbor rupture degree (ENR) and the edge scattering number (es) are calculated for the some specific graph structures. Finally, the relationships between these parameters are given.