Networks are known to be prone to node or link failures. A central issue in the analysis of networks is the assessment of their stability and reliability. The main aim is to understand, predict, and possibly even control the behavior of a networked system under attacks or disfunctions of any type. A central concept that is used to assess stability and robustness of the performance of a network under failures is that of vulnerability. A network is usually represented by an undirected simple graph where vertices represent processors and edges represent links between processors. Different approaches to properly define a measure for graph vulnerability has been proposed so far. In this paper, we study the vulnerability of cycles and related graphs to the failure of individual vertices, using a measure called residual closeness which provides a more sensitive characterization of the graph than some other well-known vulnerability measures.

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.

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