Super 3-restricted Edge Connectivity Of Direct Product Graphs

In this article, we study the super restricted edge connectivity of direct product of regular graphs. An edge cut S of G is called m-restricted if G-S contains no components of order less than m. The sizeλm(G) of minimum m-restricted edge cuts of graph G is called its m-restricted edge connectivity. Letξm(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,λm(G)≤ξm (G) holds for graph G that contains m-restricted edge cuts. Graph G is called maximally m-restricted edge connected ifλm(G)=ξm(G), and super m-restricted edge connected if any minimum m-restricted edge cut separates a component of order m. we studied edge connectivity of direct graphs and super edge connectivity of direct graphs and restricted edge connectivity of direct graphs of two regular graphs.In this article,we study three chapter.Chapter l,we study some ditinition.Chapter 2,we study Some lemmas and one theorem;Chapter 3,we study one lemma and one theorem. Let G1×G2 denote the direct product of graphs G1 and G2, letβ(G)=min{|S|:S(?)E(G) and G-S is a bipartite graph}. In this thesis, we obtain the following results:Theorem 2.2.1 If Gi is a super restricted edge connected ki-regular graph with ki≥6,2β(Gi)>3ki-2 and g(Gi)=3; i=1,2, then G1×G2 is super 3-restricted edge connected.Theorem 3.2.1 If Gi is a super restricted edge connected ki-regular graph with ki≥6,2β(Gi)>3ki-2 and g(Gi)≥4,i=1,2, then G1×G2 is super 3-restricted edge connected.