A central issue in the analysis of complex networks is the assessment of their robustness and vulnerability. A variety of measures have been proposed in the literature to quantify the robustness of networks and a number of graph-theoretic parameters have been used to derive formulas for calculating network reliability. In this paper, we study the vulnerability of interconnection networks to the failure of individual nodes, using a graph-theoretic concept of residual closeness as a measure of network robustness which provides a much fuller characterization of the network.
For a vertex v of a graph G = (V,E), the independent domination number (also called the lower independence number) iv(G) of G relative to v is the minimum cardinality of a maximal independent set in G that contains v. The average lower independence number of G is [Formula: see text]. In this paper, this parameter is defined and examined, also the average lower independence number of gear graphs is considered. Then, an algorithm for the average lower independence number of any graph is offered.
An exponential dominating set of graph [Formula: see text] is a kind of distance domination subset [Formula: see text] such that [Formula: see text], [Formula: see text], where [Formula: see text] is the length of a shortest path in [Formula: see text] if such a path exists, and [Formula: see text] otherwise. The minimum exponential domination number, [Formula: see text] is the smallest cardinality of an exponential dominating set. The minimum exponential domination number, [Formula: see text] can be decreased or increased by removal of some vertices from [Formula: see text]. In this paper, we investigate of this phenomenon which is referred to critical and stability in graphs.
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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