On Conditional Edge Connectedness Of Transitive Graphs

With the rapid development of information networks, many theoreticalproblems come into focus, one of which is the reliability of the network, thatis, the ability of the network to function even when some vertices and/or edgesfail. The underlying topology of a network is often modeled as a graph. So,some classical notations of graph theory, such as the connectivity and theedge-connectivity, is utilized to measure the reliability of networks. For fur-ther study, many variations have been introduced, which are known as higherconnectedness, such as super-κ(super-λ), hyper-κ(hyper-λ),restricted edgeconnectivity, super-restricted edge connectedness and cyclic edge connectivity,etc. In this paper, we mainly study the super-restricted edge connectednessof transitive graphs and the cyclic edge connectedness of regular half vertextransitive graphs.In chapter 1, we introduce the background of our study and some nota-tions, and recall the study of various kinds of edge connectivity. In chapter 2,we study the super-restricted edge connectedness of vertex transitive graphs,and present su?cient and necessary conditions for connected vertex transi-tive graphs to be super-λ'. In particular, super-λ'connected vertex transitivegraphs with girth g > 3 are completely characterized. In chapter 3, we studythe super-restricted edge connectedness of edge transitive graphs, and mainlycharacterize super-λ'connected edge transitive graphs. In chapter 4, we studythe cyclic edge connectedness of regular half vertex transitive graphs, we showthat a connected k(≥4)-regular half vertex transitive graph X with girthg≥6 isλ_c-optimal, and we also obtain a su?cient and necessary conditionfor a connected k(≥4)-regular half vertex transitive graph X with girth 4 tobeλ_c-optimal.In this paper,λ'-superatoms,λ_c-atoms and their non-intersection are thekey for our argumentation. The main method for proving our results is bycontradiction.