On the vulnerability concept of some graph structures

The vulnerability of a communication network measures the resistance of a network to a disruption in operation after the failure of certain processors or communication links that represent the connection between the processors. In daily life, graph that is one of the mathematical modeling method, is applied for solution of many problems. Hence, reducing complex structures to an easier model, solution is investigated. If the network is modeled by a graph, then many graph parameters like connectivity number, integrity number, toughness, scattering number, rupture, domination number can be used to calculate the vulnerability value of the network. In this thesis, the exponential domination number which is defined by Dankelmann and his friends, one of the measures of vulnerability has been computed for total graphs, complement prisms, some graphs which contain cycle and trees. Minimum exponential domination number has been investigated under graph operations including join, corona, neighborhood corona, edge corona, power and general conclusions have been obtained. Finally, for some general graphs $G$, $\gamma_{e}^{+}(G)$ and $\gamma_{e}^{-}(G)$ values have been calculated to investigate the stability of exponential domination parameter. Keywords: Graphs, Vulnerability, Domination, Exponential domination, Stability of exponential domination.