Vulnerability measures and graphs

Networks and complex systems are used to model different systems such as chemical systems, neural networks, social networks or the Internet. The study of the topology of communication networks has been growing in recent years and attracting great attention. It is important for a communication network to perform its function. Disruptions to some processors or lines of a network cause a decrease in network efficiency. This means that the network cannot perform its function. Vulnerability is an important concept used to assess the continuity and robustness of the global behavior of the network as a result of external attacks. One of the most powerful mathematical tools to examination and analyze the architecture of a network is graph theory. Thus, the vulnerability of a graph G to a network instead of the vulnerability of a network can be investigated. In determining the vulnerability of a graph, measurements such as number of dependence, durability value, integrity value, number of dominance, total number of dominance and independent saturation number are defined in the literature. In this thesis, in 2007, Arumugam and Subramanian discussed the number of independence saturation defined as a new vulnerability parameter. The independent saturation number of the G=(V, E) graph is defined as min⁡{IS(v) ∶v∈V}, where IS(v) is the largest number of elements of an independent set containing the v vertex. In this thesis, first of all, studies on this subject have been included. Subsequently, independent saturation values of some special tree graphs, E_p^t, C_((m,0)) P_n, S(a,b), B_n and DC(n,a,b) were calculated. Finally, a graph is given the definition of a splitting graph and independent saturation values of the splitting graphs are obtained for some known graphs such as and . Keywords: Graph Theory, Vulnerability, Domination, Independence Saturation, Splitting Graph