Vulnerability is an important concept in network analysis related with the ability of the network to avoid intentional attacks or disruption when a failure is produced in some of its components. Often enough, the network is modeled as an undirected and unweighted graph in which vertices represent the processing elements and edges represent the communication channel between them. Different measures for graph vulnerability have been introduced so far to study different aspects of the graph behavior after removal of vertices or links such as connectivity, toughness, scattering number, binding number and integrity. In this paper, we consider residual closeness which is a new characteristic for graph vulnerability. Residual closeness is a more sensitive vulnerability measure than the other measures of vulnerability. We obtain exact values for closeness, vertex residual closeness (VRC) and normalized vertex residual closeness (NVRC) for some wheel related graphs namely helm and sunflower.