On The Edge-connectivity Of Graphs

With the rapid development of information networks, many theoretical problems come into focus, one of which is the reliability of the network, that is, the ability of the network still function when some vertices or edges fail. The underly-ing topology of a network is often modeled as a graph. Thus, some classical param-eters of graph theory, such as the connectivity and the edge-connectivity, is used to measure the reliability of networks. For further study, many variations have been in-troduced, such as super edge-connectivity, restricted edge-connectivity, super restricted edge-connectivity, etc. In this thesis, we mainly study the edge-connectivity of several kinds of graphs.In Chapter 1, we introduce some notations and results, and review the background of the research on the edge-connectiviy.In Chapter 2, we study the relations among several kinds of edge-connectivity. At first, we give the necessary and sufficient conditions for graphs which are super-λ' and which are super-λ. Finally, we make a characterization of all graphs which areλ'-graphs but not super-λ' graphs.In Chapter 3, we study the edge-connectivity properties of bi-transitive graphs and obtain the following results:(1) Restricted edge connectivity of bi-transitive graphs equals its minimum edge degree, that is, bi-transitive graphs areλ-optimal;(2) bi-transitive graphs are super-λif and only if its degree is greater than 2;(3) We give a characterization of non-super-λ'bi-transtive graphs with degree k;(4) Using the above results, we prove that two kinds of bi-transitive graphs are super-λ'graphs.