In this paper, we study exact wormhole solutions in the framework of general relativity with a general equation of state that reduced to a linear equation of state asymptotically. By considering a special shape function, we find classes of solutions which are asymptotically flat. We study the violation of NEC as the main ingredient in the wormhole physics. We investigate the possibility of finding wormhole solutions with asymptotically different state parameter. We show that in principle, wormhole with a vanishing redshift function and the selected shape function, cannot satisfy NEC at large distance from wormhole. We present solutions which have the positive total amount of mater in the "volume integral quantifier" method. For this class of solutions, fluid near the wormhole throat is in the phantom regime and at some r = r2, the phantom regime is connected to a dark energy regime. Thus, we need small amount of exotic matter to construct wormhole solutions.

This paper discusses wormholes supported by general equation-of-state , resulting in a significant combination of the linear equation-of-state and some other models. Wormhole with a quadratic equation-of-state is studied as a particular example. It is shown that the violation of null energy condition is restricted to some regions in the vicinity of the throat. The combination of barotropic and polytropic equation-of-state has been studied. We consider fluid near the wormhole throat in an exotic regime which at some r = r 1 , the exotic regime is connected to a distribution of asymptotically dark energy regime with −1 < ω < −1/3. We have presented wormhole solutions with small amount of exotic matter. We have shown that using different forms of equationof-state has a considerable effect on the minimizing violation of the null energy condition. The effect of many parameters such as redshift as detected by a distant observer and energy density at the throat on the r 1 is investigated. The solutions are asymptotically flat and compatible with presently available observational data at the large cosmic scale.

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