ON THE RUPTURE DEGREE OF A GRAPH

Abstract

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 ⊂ 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.