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.

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.

A graph or molecular graph’s overall resilience and connectedness are gauged by the Sombor Index (SOI). It is an improved version of the conventional SOI that is designed to capture ambiguity in the relationships between nodes or atoms. This indicator aids in assessing the network’s resilience and stability in erratic circumstances. We have investigated the charac-teristics of the SOI of neutrosophic graphs (N Gs) in this study. The association between the neutrosophic first Zagreb Index (N FZI) of N Gs and the neutrosophic Sombor Index (N SI) was revealed in the study. The application of SOI based on Site Selection for Thermal Power Plants in N Gs is finally covered.

This paper introduces the concept of the direct product of sets that involve Fermatean neutrosophic elements in structures called INK-algebras. It defines terms like the direct product of Fermatean neutrosophic INK-ideals in INK-algebras and Fermatean neutrosophic sets (FNSs), Fermatean neutrosophic INK-ideals (FNINK-Is), and Fermatean neutrosophic closed INK-ideals (FNCINK-Is). The proof of theorems illustrating the relationships between these ideas is included in the paper. It also defines the INK-sub algebra embedded in an INK-algebra and gives a theorem elucidating the connection between the direct product of Fermatean neutrosophic INK-ideals and the images of these sub-algebras. In essence, the paper investigates and establishes connections between different mathematical ideas concerning INK-algebras and FNSs.

This study introduces the notion of complex t-neutrosophic graphs (CTNGs) as a powerful tool for understanding and displaying complex interactions that are sometimes difficult to understand. It demonstrates that CTNGs may accurately reflect complicated interactions involving several components or dimensions within a particular scenario. It also instructs the basic set operations of CTNGs and analyzes notions like homomorphism and isomorphism within this framework. Furthermore, the research describes a practical application of CTNGs. It illustrates their value in addressing biodiversity conservation by taking into account a variety of relevant factors. The paper uses this application to highlight the flexibility and effectiveness of CTNGs as a tool for decision-makers to visualize and prioritize activities targeted at improving biodiversity conservation.

Dealing with uncertainty and imprecision is critical in modern decision-making processes, particularly in multi-criteria situations. Exponential Intuitionistic Fuzzy Sets (EIFS) are a sophisticated mathematical framework that extends intuitionistic fuzzy sets by integrating an exponential function, improving its capacity to adequately capture ambiguity. This study investigates several EIFS operations, such as EIF intersection and union, which expand conventional set operations to accommodate partial and non membership NM degrees more flexibly. Furthermore, we look into crucial operations including simple difference, limited difference, disjoint sum, disjunctive sum, and Cartesian product, all of which play important roles in modelling uncertainty in decision processes. A case study on raw material purchase demonstrates how these processes might be applied. The decision-making system takes into account numerous providers, analysing aspects such as cost, quality, and delivery time with EIFS-based aggregation approaches. Using these activities, decision-makers may improve supplier performance, reduce risks, and optimise procurement strategies in unpredictable circumstances.