Graph Connectivity Parameters

In the process of designing and guarding computers or communication networks, the vulnerability of them is an important performance criterion. Therefore, one of the basic ideas to design networks is that they do not easily get disrupted under external attack and, these are easily reconstructed if they do really get disrupted. We often modeled the networks by connected graphs, where the communication stations are represented by the vertices and the direct communication links are represented by the edges. Under this mathematical model of networks, people began to find some criteria to measure the vulnerability of networks. At the beginning, vertex-connectivity and edge-connectivity are two parameters often used for it. But recently, people found that these two parameters do not do very well for it, so some new parameters, such as toughness, scattering number, integrity, tenacity have been introduced in the past decade. All of these parameters are more reasonable than vertex-connectivity and edge-connectivity, for they can measure not only the amount of work done to damage the network but also how badly the network is damaged.In this thesis, by summing up and comparing of the main results about these parameters, we introduce a new connected parameter, the relative tenacity of graphs, which is convenient and objective to measure the vulnerability of networks in some extent, and do some preliminary research on it. In Chapter 1, we analyze the significance for discussing the connectivity parameters, then present the main results in this thesis. In Chapter 2, we survey the results of the connectivity parameters (toughness, scattering number, integrity, tenacity). In Chapter 3, we determine the relative tenacity of some specific families of graphs (such as complete graphs, cycles, paths, stars, comets, complete bipartite graphs and complete k -partite graphs). The bounds of the relative tenacity, the relative tenacity of the operation of graphs, the relationship between it and other parameters, the relative tenacity and it's algorithm for trees are also investigated in this chapter. In Chapter 4, the maximum and minimum networks with givingorder and relative tenacity are also characterized. In Chapter 5, we discuss the edge-relative tenacity of graphs, and in Chapter 6 we discuss the computation of the relative tenacity of graphs. Finally, some ideas are proposed for further study of the relative tenacity of graphs.