In a communication network, vulnerability measures the resistance of the network to disruption of operation after the failure of certain stations or communication links. If we think of a connected graph as model-ing a network, the rupture degree of a graph is one measure of graph vulnerability and it is defined by
r(G) = max{w(G-S)-|S|-m(G-S): S \subset V(G), w(G-S)>1}
where w(G-S) is the number of components of G-S and m(G-S) is the order of a largest component of G-S. In this paper, general results on the rupture degree of a graph are considered. Firstly, some bounds on the rupture degree are given. Further, rupture degree of a complete k-ary tree is calculated. Also several results are given about complete k-ary tree and graph operations. Finally, we give formulas for the rupture degree of the cartesian product of some special graphs.