Sucient Conditions For A Graph To Be Super 3-restricted Connected With Given Girth

Numerous networks as, for example, transport networks, road networks, electrical net-works, telecommunication systems or networks of servers can be modeled by graphs.The fault tolerance of the network, that is, the ability of the network to function evenwhen some vertices or edges fail, has come into focus. The traditional connectivity usedto measure the reliability have many deficiencies. Thus, many conceptions of higherconnectedness have been introduced, such as super-κm, super-λm, m-restricted vertex(edge) connectivity, etc, where m is integer. In this thesis, we study the super-κ3,λ3-optimal and super-λ3 of graph with given girth.In chapter 1, we introduce the background of our study and some terminologies. Inchapter 2, we study theλ3-optimality in graphs with given girth and prove that if Gisλ3 connected graph with girth g≥7, minimum degreeδ≥3 and diameter D≤g ? 3,then G isλ3-optimal. Furthermore, we give two su?cient conditions for a graph to beλ3-optimal with diameter D = g ? 2. In chapter 3, we study super-λ3 graphs with givengirth and prove that if G is connected graph with girth g≥7, minimum degreeδ≥3and diameter D≤g ? 5, then G is super-κ3, and if D≤g ? 4 then G is super-λ₃.