Connectedness Of Bi-Cayley Graphs And Bi-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 modelled 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) and r-restricted edge connectivity,etc.In chapter 1, we introduce the background of our study and some notations,and give definitions of Bi-Cayley graphs and Bi-transitive graphs. In chapter2, we study connectivity and prove that all connected Bi-Cayley graphs andBi-transitive graphs are optimally ?κ. In chapter 3, we study super connect-edness of two kinds of graphs. we prove that a connected vertex transitivebipartite graph is not super ?κif and only if it is isomorphic to the lexico-graphic product of a cycle Cn(n≥6) by a null graph Nm, and characterizenon-hyper ?κconnected vertex transitive bipartite graphs. In addition, wecharacterize non-hyper?κconnected Bi-transitive graphs, and derive two suf-ficient conditions for connected Bi-transitive graphs to be super?κ. In chapter4, we study higher order edge connectedness and obtain that: (1) all connectedBi-transitive graphs are super ?λ; (2) all connected k-regular Bi-transitivegraphs with at least 4 vertices are optimally-λ; (3) a necessary and su?cientcondition is gived for connected Bi-Cayley graphs to be optimally-λ(3).In the dissertation, various atoms and their non-intersection are the keyfor our argumentation. The main method for proving our results is apagoge.