The Average Lower Connectivity of Graphs

Abstract

For a vertex $v$ of a graph $G$, the lower connectivity, denoted by $s_v(G)$, is the smallest number of vertices that contains $v$ and those vertices whose deletion from $G$ produces a disconnected or a trivial graph. The average lower connectivity denoted by ${\kappa }_{\text{a}\text{v}}(G)$ is the value $({\sum }_{v\in V\left(G\right)}{s}_v\left(G\right))/\left|V\left(G\right)\right|$. It is shown that this parameter can be used to measure the vulnerability of networks. This paper contains results on bounds for the average lower connectivity and obtains the average lower connectivity of some graphs.