A mathematical method of combining several elements has emerged in recent times, providing a more comprehensive approach. Adhering to the foregoing mathematical methodology, we fuse two extremely potent methods, namely graph theory and neutrosophic sets, and present the concept of neutrosophic graphs ($\aleph G$). Next, we outline many ideas, such as union, join, and composition of $\aleph G$s, which facilitate the straightforward manipulation of $\aleph Gs$ in decision-making scenarios. We provide a few scenarios to clarify these activities. The homomorphisms of $\aleph Gs$ are also described. Lastly, understanding neutrosophic graphs and how Japan responds to earthquakes can help develop more resilient and adaptable disaster management plans, which can eventually save lives and lessen the effects of seismic disasters. With the support of using an absolute score function value, Hokkaido (H) and Saitama (SA) were the optimized locations. Because of its location in the Pacific Ring of Fire, Japan is vulnerable to regular earthquakes. As such, it is critical to customize reaction plans to the unique difficulties and features of Japan’s seismic activity. Examining neutrosophic graphs within the framework of earthquake response centers might offer valuable perspectives on tailoring and enhancing response tactics, particularly for Japan’s requirements.

Complex neutrosophic graphs are created by combining complex neutrosophic sets with graph theory ideas. This provides a flexible framework for tackling complex problem-solving circumstances. Various processes, such as union, join, and composition, are thoroughly investigated to improve the administration of complicated neutrosophic graphs. This research also looks into the area of complicated neutrosophic graph homeomorphisms, investigating the transformations and mappings that occur inside these structures. This investigation advances our knowledge of the intrinsic features and linkages found in complicated neutrosophic graphs. Finally, our methodology’s practical usefulness is demonstrated by its use in the design of hospital infrastructure. We illustrate the effectiveness of complicated neutrosophic graphs in addressing difficult design concerns and optimizing infrastructure solutions for practical deployment in this real-world scenario.

Production systems prioritize economic efficiency, often overlooking environmental sustainability. Growing environmental awareness has led to circular production systems that integrate economic and ecological goals. This study presents a sustainable production–inventory model for a two-echelon supply chain with a manufacturer and retailer, incorporating remanufacturing under carbon emissions regulations. The manufacturing process follows a price and time-dependent demand while remanufacturing relies on a circularity index demand. Two model variants are analyzed using Hessian algorithms, combining cap-and-trade and carbon tax policies with different demand rates and profit functions. Results show that maximum profit occurs under cap-and-trade, despite an 82.20% circularity index. Carbon regulations effectively reduce emissions but slightly lower profitability. This study highlights the need to optimize inventory and remanufacturing decisions to balance economic and environmental goals within a circular economy.

The main concept involved in this study explains the theme of complex t-intuitionistic fuzzy graphs (CTIFGs), which act as a powerful tool in analyzing and displaying the relationships among various applications that are difficult to recognize. The manuscript also demonstrates the capability of CTIFGs to create complex associations with multiple domains when considering a physical situation. Following this, the basic set of operations for CTIFGs is projected. The ideas on isomorphism and homomorphism of the CTIFGs are also presented. Moreover, the manuscript describes the importance of the above-mentioned technique in an effective way, giving a solution to the practical application associated with rubber processing industrial wastewater. The contributing factors and corresponding interdependencies are considered when calibrating the complex nature of industrial wastewater associated with the CTIFGs. The results highlight the adaptability and possible efficiencies of CTIFGs, which act as a decision-making tool and also indicate their importance for policy planners in important societal issues.

Trapezoidal-valued fermatean fuzzy numbers (TpVFFNs) are essential for handling daily decision-making issues in the engineering and management fields. Accumulation processes on the set of TpVFFN are used to address decision-making problems described in this environment as necessary. The primary goal of this paper is to provide the concept of Dombi t-norm $\left(D_{tn}\right)$- and Dombi t-conorm $\left(D_{tcn}\right)$-based accumulation operators on the class of TpVFFN, emphasizing how they behave symmetrically in aggregation processes to maintain consistency and fairness. To use s to illustrate mathematical circumstances, we first create a trapezoidal-valued fermatean fuzzy Dombi’s weighted geometric operator, hexagonal hybird geometric operator, fermatean fuzzy order weighted geometric operator. Second, we use a multi-attribute group decision-making (MAGDM) approach to compute the recommended accumulation operators. Finally, we demonstrate the potential practical application of the proposed decision-making problem related to the pink cab.

This work introduces the notion of a hesitant bipolar-valued intuitionistic fuzzy graph (HBVIFG), which reflects four different characterizations: membership with positive/negative aspects and non-membership with positive/negative aspects, incorporating multi-dimensional alternatives in all of its information. HBVIFG generalizes both HBVFG and BVHFG due to its diversified nature in observing four perspectives along with multiple attributes in a piece of information. Numerous studies, examples, and graphical representations emphasize the concept’s distinctiveness and importance. The following graph theory terms are defined: strong directed HBVIFG, full directed HBVIFG, directed spanning HBVIFSG, directed HBVIFSG, and partial directed hesitant bipolar-valued intuitionistic fuzzy subgraph (HBVIFSG). Examples of operations utilizing two HBVIFGs are Cartesian, direct, lexicographical, and strong products. A scenario is used to generate the mapping of relations, which includes homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism. We describe a directed HBVIFG application that employs an algorithm to determine the most dominant person and self-persistent person in a social system and a comparative study is also provided. The proposed method provides a more detailed framework for assessing the most dominant and self-persistent individual in a social network across multi-level attributes along with positive and negative side membership and non-membership grades in each element of a network.