Study On The Stability Of The Network Parameters

In design and construction of a network, stability is an important factor that must be taken into consideration. A network can be modeled by a connected graph, in which vertices represent modules and edges represent channels between the modules. Therefore the research of connectivity and stability of graphs has been a hot point in graph theory for long time. Centering on this topic, except connectivity and edge-connectivity in early period, some new connected parameters of graph were introduced in succession. They are binding number, scattering number, integrity and edge-integrity, tenacity and edge-tenacity, toughness etc. these parameters show the most serious damage and the situation of a network after the damage simultaneously, so they measure the connectivity of graphs more accurate.While consider the stability of some special networks, for example, the spy network, the difference is obviously. If one spy is captured, the adjacent spies will be useless to the network as a whole. With the background stated above, G..Gunther introduced vertex-neighbor-connectivity in 1991. In 1994 and 1996, S.S. Y. Wu and M.B. Cozzens introduced vertex-neighbor-integrity and edge-neighbor-integrity. Consequently, a series of related results are emerged. These results improve the study of stability theory of networks greatly. Based on these achievements, this paper mainly considers the stability of networks with neighbor condition. Some results are obtained, therefore the stability theory of graphs are enriched in some extent.The main contents are showed as followsAs an introduction, Chapter 1 introduces present research situation of connectivity and stability of graphs, and points out the background and significance of the work in this thesis.Chapter 2 focuses on the discussion of vertex-neighbor-integrity of compound graphs. We give several upper and lower bounds of the vertex-neighbor-integrity of compound graphs in general condition. Meanwhile, the vertex-neighbor-integrity of some special compound graphs such as path, cycle etc. are given. At last, we correct several mistakes in a paper written by Pinar Dundar.In Chapter 3, the concept of vertex-neighbor-scattering number of graphs is introduced, and the values of vertex-neighbor-scattering number of some simple graphs are given. Furthermore, the vertex-neighbor-scattering number of trees is discussed in details.The main contents of Chapter 4 are the introduction of edge-neighbor-scattering number of graphs, the edge-neighbor-scattering number values of several simple graphs and a polynomial algorithm of the edge-neighbor-scattering number of trees.