The Rupture Degree and Gear Graphs

n a communication network, several vulnerability measures are used to determine the resistance of the network to disruption of operation after the failure of certain stations or communication links. If we think of a graph as modelling 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 of 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 we give some results on the rupture degree of gear graphs. Also the relationships between the rupture degree and some vulnerability parameters, namely the tenacity and toughness, are given.