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Economic development and environmental sustainability have recently become global priorities. This study utilizes the nonlinear autoregressive distribution lag model (NARDL) to investigate the asymmetric effects of natural resource rents on economic growth and environmental quality in China and the United States from 1970 to 2020. The findings indicate that natural resource rents exert asymmetric effects on economic growth and environmental quality in both countries; however, the impact patterns differ between the two countries. In the short term, a decrease in natural resource rents significantly increases GDP per capita in the US, while an increase in natural resource rents has no effect. Moreover, natural resource rents do not influence the ecological footprint in the US. By contrast, China's GDP per capita remains unaffected by natural resource rents, while the ecological footprint is significantly influenced by both positive and negative changes in natural resource rents, exhibiting asymmetric impacts. In the long run, an increase in natural resource rents significantly impacts both GDP per capita and the ecological footprint in the US, while a decrease only impacts the ecological footprint. By contrast, both increases and decreases in natural resource rents impact per capita GDP and the ecological footprint in China. These findings highlight the importance of tailoring country‐specific policies to facilitate sustainable economic development and enhance environmental quality. [ABSTRACT FROM AUTHOR]

The Fermat problem is a crucial topological issue corresponding to fractal networks. In this paper, we discuss the average Fermat distance (AFD) of the Vicsek polygon network and analyze structural properties. We construct the Vicsek polygon network based on Vicsek fractal in an iterative way. Given the structure of network, we present an elaborate analysis of the Fermat point under various situations. The special network structure allows a way to calculate the AFD based on average geodesic distance (AGD). Moreover, we introduce the Vicsek polygon fractal and calculate its AGD and AFD. Its relationship with the network enables us to deduce the above two indices of the network directly. The results show that both in network and fractal, the ratio of AFD and AGD tends to 3/2, which demonstrates that both of them can serve as indicators of small-world property of complex networks. In fact, in Vicsek polygon network, the AFD grows linearly with network order, implying that our evolving network does not possess the small-world property. [ABSTRACT FROM AUTHOR]