Edge Connectivity and Super Edge-Connectivity of Jump Graphs

For a graph G, let (G) over bar and L(G) denote the complement graph and the line graph of G, and let (G) over bar denote the jump graph of G. It is well-known that (G) (G) (G). A graph G is called maximally edge connected if G has edge connectivity of its minimum degree and is called super-(edge-)connected if for every minimum vertex(edge) cut S of G, G - S has isolated vertices. In this paper, we apply previous results of edge-connectivity and super-connectivity and super-edge-connectivity to give the following four results on jump graphs: (1) Characterization of the graph G having (J (G)) = 2; (2) If J(G) is connected, then J(G) is maximally edge connected; (3) Characterization of the graph G whose J(G) is super edge connected but not super-connected; (4) Give a sufficient condition for J(G) is super edge connected.