Rupture Degree and Weak Rupture Degree of Gear Graphs

Abstract

Various vulnerability parameters were defined in graph theory to study
the vulnerability of the networks. In a communication network, several vulnerability measures are used to determine the resistance of the network to distruption of operation after the failure of certain stations or communication links. If we consider a graph as a model of a communication network, the rupture degree and weak rupture degree of a graph are the measures of graph vulnerability. In this paper, we determine the exact values of the rupture degree of $G_n$ and $G_m \Box G_n,$ where $m,n \geq 4$ and weak rupture degree of $G_n,$ where $n\geq 4,$ $\overline G_n,$ where $n\geq 4,$ $L(G_n)$ where $n\geq 4,$ $K_2 \Box G_n,$ where $n\geq 4$ and $G_n \Box G_m,$ where $n,m \geq 4.$