Edge Connectivity Of Strong Product Graphs And The Limit Of The Product Graph Of The Dictionary

In this article,we study the restricted edge connectivity of strong product and lexicographic product of regular graphs.An edge cut S of a connected graph G is called m-restricted if G-S contains no components of order less than m.The sizeλ_m(G) of minimum m-restricted edge cuts of graph G is called its m-restricted edge connectivity.Letζ_m(G) denote the minimum size of sets of edges with exactly one end in any given connected vertex-induced subgraph of order m.It is known that when m≤3,λ_m(G)≤ζ_m(G) holds for graph G that contains m-restricted edge cuts.Graph G is called maximally m-restricted edge connected ifλ_m(G)=ζ_m(G),and super m-restricted edge connected if any minimum m-restricted edge cut separates a component of order m.In this thesis,we obtain the following results.Theorem2.1.7 If G₁ and G₂ are super edge connected regular graphs with degree at least two,then G₁(?) G₂ is supper-λ.Theorem2.2.3 If G₁ and G₂ are maximally edge connected regular graphs with degree at least two,then G₁(?) G₂ is supper restricted edge connectedTheorem3.2.1 Let G_i be k_i-regular graphs with k_i≥2,i=1,2.Ifλ(G₁)=k₁ andλ(G₂)=k₂ -1,then G₁ o G₂ is maximal restricted edge connected.Theorem3.3.3 Let G₁ and G₂ be two regular graphs with degree at least two. If they are maximally edge connected,then G₁ o G₂ is super restricted edge connected.