are $\Omega_{dm}^0=0.25$, $\Omega_X^0=0.7$, and $\Omega_b^0=0.05$. In Fig.1, we show the cases having the common equation-of-state parameter $w_X=-1.2$, while the interaction parameters $\alpha$ are taken to be $0$, $0.5$, and $0.8$, respectively. Note that the $\alpha=0$ case corresponds to a normal phantom model with constant $w_X$. In this example we see the role the interaction parameter $\alpha$ plays in the model. The transfer energy flows from dark matter to dark energy when $\alpha>0$; the larger $\alpha$ leads to the stronger energy flow; density of baryon component is also affected evidently by the interaction between dark energy and dark matter. In Fig.2, we depict the cases with common interaction parameter $\alpha=0.5$, and the equation-of-state parameters $w_X$ are taken to be $-1$, $-0.8$, and $-1.2$, respectively. Here $w_X=-1$ case corresponds exactly to the GCG model. The effect of the parameter $w_X$ in the NGCG scenario is also evident as we see in this example.

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Let us now discuss the cosmological consequences led by the NGCG model and compare the cosmological quantities in the NGCG cosmology with those of some special cases such as $\Lambda$CDM and GCG. Firstly, we regard the Hubble parameter $H$ which evaluates the expansion rate of the Universe. 