The vulnerability of a network measures 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 modeling a network, several vulnerability measures have been used to describe the stability of networks, including connectivity, toughness, scattering number, binding number and integrity. We consider a new characteristic, residual closeness which is more sensitive than the well-known vulnerability measures. Residual closeness measures the network resistance evaluating closeness after removal of vertices or links. In this paper, closeness, vertex residual closeness (VRC) and normalized vertex residual closeness (NVRC) of wheels and some related networks namely gear and friendship graph are calculated, and exact values are obtained.