Triangular Diagram Restricted Edge Connectivity

The edge connectivity of a graph is the tradition&l measurement of the fault tolerance of a network. But there are some shortages when it measures the fault tolerance of a network.Esfanian and Hakimi put forward a concept named the restricted edge connectiv-ity to make up the shortages.In1994,Fabrega and Fiol promoted the concept of the restricted edge connectivity,they put forward κ-restricted edge connectivity.The first chapter introduces the graph-theoretical terminology.The second chapter introduces how the problem put forward and the present situation of the problem.The third chapter discusses the minimum edge degree conditions for triangle-free graphs connectivity to be λ'-optimal and super-λ'.In the first section we prove that(a)Let G be a connected triangle-free graph of order n and minimum degree δ(G)≥2.If the minimum edge degree ξ(G)≥(n/2-2)(1+1/(δ(G)-1)),then G is λ'-optimal.By the result,a corollary is deduced(b)Let G be a connected triangle-free graph of order n.If the minimum degree δ(G)≥n/4+1,then G is λ'-optimal.In the second section,we prove that(c)Let G be a connected triangle-free graph of order n and minimum edge degre δ(G)≥3.If the minimum edge degree ξ(G)(n/2-2)(1+1/(δ(G)-2))+1,then G is Super-λ'.Then a corollary is given(d)Let G be a connected triangle-free graph of order n.If the minimum degree δ(G)≥「(n+2)/4」+2,then G is super-λ'.The forth chapter studies the degree sum conditions for triangle-free graphs connec-tivity to be Super-λ'.We prove that(e)Let G be a connected triangle-free graph of order n≥4.Ifd(u)+d(v)≥2「(n+2)/4」+3,for each pair u,v∈V(G) with dist(u,v)=2,then is super-λ'.Then a corollary is deduced(f)Let G be a connected bipartite graph of order n≥4.If d(u)+d(v)≥2「(n+2)/4」+3,for each pair u,v∈V(G)with dist(u,v)=2,then G is super-λ'.The fifth chapter studies the minimum edge degree conditions for bipartite graphs connectivity to be Super-λ3basing on the conclusions of the third and the forth chapter.We prove that(g)Let G(X∪Y)be a λ3-optimal connected bipartite graph,the minimum degree δ(G)≥4.U is a λ3-atom of G,and|U|≥4,If G is not super-λ3,then Then we can get a corollary(h)Let G(X∪Y)is a λ3-optimal connected bipartite graph,δ(G)≥4.If ξ3(G)≥n-1,then G is super-λ3.