| To avoid some defects of traditional edge-connectivity in measuring the reli-ability of networks, Esfahanian and Hakimi proposed the concept of restricted edge connectivity.A set F of edges of a connected graph G is said to be a restricted edge-cut, if its removal disconnects G, and every connected component of G - F contains at least two vertices. Let the graph G have a restricted edge-cut, if the cardinality of every minimum restricted edge-cut of G equals to the minimum edge degree of G, then G is said to be λ’-optimal, and if every minimum restricted edge-cut of G isolates an edge, then G is said to be super-A’. In this paper, we will investigate connected super-λ’ triangle-free graphs (’triangle-free’ means the graph contains no 3-cycles).The first chapter will introduce some basic concepts and background regarding graphs and networks.The second chapter will introduce some definitions and some research topics related to triangle-free graphs.The third chapter will give some sufficient conditions for connected triangle-free graph being A’-potimal and super-λ’. Let G be a connected triangle-free graph of order at least 4 and let T(G)= min{d(u)+d(v)|u,v ∈ G,dist(u, v)= 2}, It is proved that if T(G)≥ 2[n+2/4]+3, then the graph G is super-λ’. This improves the following result [2]:if T(G)≥ 2[n+2/4]+1, then the graph G is λ’-optimal.The forth chapter will analyze the structure of A’-superatoms of a triangle-free graph G under certain conditions. Let G is a connected triangle-free graph of order at least 4 with T(G)≥ 2[n+2/4]+1. If X is a A’-superatom of G, then G[X] is isomorphic to one of the graphs: Km,m, Km,m+1, or Km,m+1 - K2, and moreover, if G[X] is Km,m+1 - K2, then [X, X] consists of 2m - 1 independent edges. Furthermore, a characterization is given of all nonsuper-λ’ triangle-free graphs with T(G)≥ 2[n+2/4]+1. |
