Edge vulnerability parameters of split graphs

A graph is called a split graph   if its vertex set can be partitioned into a clique and an independent set. In this work, we investigate three vulnerability parameters of split graphs when edges are removed, i.e., edge-connectivity, edge-toughness and edge-integrity. It is proved that, for a noncomplete connected split graph GG, its edge-connectivity is δ(G)δ(G), and its edge-toughness is min{δ(G),|E(G)||V(G)|−1}, where δ(G)δ(G), V(G)V(G) and E(G)E(G), are the minimum degree, the vertex set and the edge set of GG, respectively. Furthermore, we show that the edge-integrity of a noncomplete connected split graph equals its order when its minimum degree is greater than half of the size of its largest clique.