Graphs and scattering number

In a communication network, the vulnerability is the resistance of the network to disruption of operation after the failure of certain stations or communication links. A communication network can be modelled by a graph as stations corresponding to the vertices of the graph and communication links corresponding to the edges of the graph. There are many measurement such as Connectivity, Toughness, Scattering Number, Integrity, Rupture Degree defined over graphs to measure the vulnerability of networks. In the first section, firstly the concept of vulnerability is considered, then some basic definitions needed for the thesis are given. Then, the definitions of vulnerability measurements mentioned above and some results in the literature related these measurments are given. After, an example is given about the scattering number and some results in the literature are listed. In the second section, the scattering number of some proper graph products is studied. Firstly, the graph of K1,m x K1,n (m≥2, n≥2) is examined and the scattering number of this graph is given. After, the graphs K1,m x Pn and K1,m x Cn (m≥2, n≥2) are investigated and scattering numbers of these graphs are obtained. Finally, the scattering number of graph K2 x Cn is calculated. In the last section, the results obtained in the second section are compared and the relationship between scattering number and independence number and covering number is studied.