On edge-rupture degree of graphs

The edge-rupture degree of an incomplete connected graph G   is defined as r′(G)=max{ω(G−S)−|S|−m(G−S):S⊆E(G),ω(G−S)>1},r′(G)=max{ω(G−S)−|S|−m(G−S):S⊆E(G),ω(G−S)>1}, where ω(G−S)ω(G−S) and m(G−S),m(G−S), respectively, denote the number of components and the order of a largest component in G−SG−S. This is a reasonable parameter to measure the vulnerability of networks, as it takes into account both the amount of work done to damage the network and how badly the network is damaged. In this paper, firstly, the relationships between the edge-rupture degree and some other graph parameters, namely the edge-connectivity, edge-integrity, edge-toughness, edge-tenacity, diameter, the algebraic connectivity and the minimum degree are established. After that, the edge-rupture degree of the middle graphs of path and cycle are given. Then, we introduced the concept of r′-maximal graph and give some basic results of such graphs. Finally, we introduce the concept of edge-ruptured and strictly edge-ruptured graph, and we establish necessary and sufficient conditions for a graph to be edge-ruptured and strictly edge-ruptured, respectively.