A Study on Complex t-Neutrosophic Graph with Intention to Preserve Biodiversity

Zadeh [1] fuzzy set theory provides a mathematical framework for addressing ambiguity, vagueness, and imprecision in computational perception. Since its origin, fuzzy set theory has been successfully applied in a variety of technological and scientific disciplines, including consumer electronics, control systems, image processing, robotics, artificial intelligence, and industrial automation. Furthermore, it has proved useful in operations research areas such as project management, decision theory, supply chain management, queuing theory, and quality control. Authors such as Kandel [2], Klir, and Yuan [3], as well as Mendel [4] and Zimmermann [5], write introductory works that explain the concepts and principles of fuzzy set theory to facilitate further comprehension and application.

The intuitionistic fuzzy set (IFS) is an extension of the fuzzy set that includes both membership and non-membership functions, expanding Zadeh’s original notion. Atanassov [6] proposed IFS as an extension of Zadeh’s fuzzy set, which in turn extends on the standard concept of a set. These sets provide a flexible framework for dealing with the uncertainty and ambiguity that are inherent in decision-making processes. De et al. [7] presented operations on IFS and investigated their varied properties, shedding light on key components of these procedures. IFS is important in fuzzy mathematics because it has a wide range of real-world applications, such as pattern identification, machine learning, decision-making, and market forecasting. Ejegwa et al. [8] offered a thorough analysis of numerous IFS models in actual circumstances. Smarandache [9] established Neutrosophy in 1998, which investigates the genesis, nature, and scope of neutralities, as well as their interconnections across many ideational spectrums. A neutrosophic set is defined by three membership functions: truth, indeterminacy, and falsity. This framework is an effective mathematical tool for extending the ideas of classical sets, fuzzy sets [1], IFSs [5], interval-valued fuzzy sets [10], paraconsistent sets, dialetheist sets, paradoxist sets, and tautological sets [9]. Neutrosophic sets are effective at managing the uncertain and inconsistent information that is omnipresent in daily life. In recent years, scholars throughout the world have conducted substantial studies on neutrosophic sets. Wang et al. [11] researched single-valued neutrosophic sets, making them more useful in scientific and technical sectors where reflecting uncertainty, incompleteness, imprecision, and inconsistency is critical. Hanafy et al. [12,13] investigated the correlation coefficient of neutrosophic sets, whereas Ye [14] looked into the correlation coefficient of single-valued neutrosophic sets. Broumi and Smaradache [15] defined the correlation coefficient for interval neutrosophic sets, whereas Salama et al. [16] investigated neutrosophic sets and topological spaces.

Graphs are a useful method to illustrate relationships between items, with edges indicating relationships and vertices representing objects. Graph theory is a strong tool for understanding and simplifying complicated systems. In mathematical chemistry, molecular descriptors are important for investigating the structure of molecules using mathematical approaches. Chemical graph theory investigates the intersection of chemistry, graph theory, and mathematics, using molecular graphs to represent atoms and bonds in compounds, allowing for the depiction of connections. Creating a Fuzzy Graph Model is necessary when expressing item connections when there is uncertainty. Fuzzy graphs are excellent mathematical tools for dealing with unknown elements. Rosenfeld [17] pioneered research in fuzzy graph theory, which was followed by Mordeson and Chang-Shyh’s [18] discussion of fuzzy graph operations and Bhattacharya’s [19] proof of graph theoretic findings. Bhutani [20] investigated the automorphisms of fuzzy graphs. Fuzzy graph theory has several applications in science and engineering, including broadcast communications, production, social networks, artificial intelligence, data processing, and neural systems. Intuitionistic fuzzy graphs (IFGs) provide a finer model of human cognition and decision-making processes, allowing for the exact determination of acceptance or rejection within a set. Shannon and Atanassov [21] introduced IFGs and intuitionistic fuzzy relations, which have been studied for their properties. Karunambigai and Atanassov [22] focused on IFG operations, whereas Gani and Begum [23] investigated IFG size, order, and degree. Sundas and Akram [24] explained how intuitionistic fuzzy soft graphs may be used to solve decision-making difficulties. Yaqoob et al. [25] contributed to the advancement of complicated intuitionistic fuzzy graph theory. Abida and Faryal [26] classified essential operations for complex IFGs into direct, semi-strong, strong, and modular products. Quek et al. [27] develop pentapartitioned neutrosophic graphs and show how they may be used to determine the safest pathways and towns in response to COVID-19. They investigate mathematical models that take neutrosophic components into account, resulting in a complete framework for assessing complex pandemic responses. AL Al-Omeri and Kaviyarasu [28] present a method for detecting online streaming services based on the Max product of complement in neutrosophic graphs. Their method uses neutrosophic graphs to model and evaluate internet streaming networks, assisting in the discovery and optimization of streaming services. Broumi et al. [29] introduce the notion of single-valued neutrosophic graphs, which expands graph theory to include neutrosophic components. Their work strengthens the theoretical underpinning of neutrosophic graph theory by providing insights into the representation and analysis of ambiguous and indeterminate data and later provides complex fermatean neutrosophic graphs and illustrates their use in decision-making processes. By incorporating fermatean neutrosophic aspects into graph theory, they present a unique paradigm for dealing with choice issues in uncertain contexts [30]. Akram and Shahzadi [31] investigate basic mathematical characteristics and procedures related to neutrosophic graph theory. Their work advances the creation of mathematical methods for assessing uncertain and indeterminate data represented by neutrosophic graphs. Yaqoob and Akram [32] define complex neutrosophic graphs, which expand the idea of neutrosophic graphs to include complex-valued neutrosophic constituents. Their work broadens the scope of neutrosophic graph theory by introducing a flexible framework for modeling and evaluating complicated systems with uncertain knowledge. Şahin’s [33] approach to neutrosophic graph theory with applications provides insights into the theoretical underpinnings and practical applications of neutrosophic graph models. Their work advances neutrosophic graph theory, making it easier to analyze complex systems in uncertain environments. Kaviyarasu’s [34] research focuses on the mathematical and structural features of neutrosophic graphs with regular edge patterns. Their study advances our understanding of neutrosophic graph topologies, providing insights into the representation and processing of ambiguous data. Mohammed Alqahtani et al. [35] investigate the use of complicated neutrosophic graphs in hospital infrastructure design. Their research shows how complicated neutrosophic graph models may be used to enhance hospital infrastructure layouts, hence increasing efficiency and resource allocation in healthcare institutions. Asima Razzaque et al. [36] explain a comprehensive analysis to reduce poverty by using t-intuitionistic fuzzy graphs. In [37,38,39,40,41] discussed some diction making by using FG, IFG, NFG and CNFGs. When compared to traditional complex neutrosophic fuzzy sets, the use of complex t-neutrosophic fuzzy sets (CTNFSs) offers significant benefits in negotiating ambiguity and uncertainty. This strategic approach provides a flexible technique for dealing with the uncertainties and ambiguities that are inherent in decision-making processes. Furthermore, complex t-neutrosophic fuzzy models are gaining popularity as a way to bridge the gap between traditional numerical models used in engineering and research and the symbolic models found in expert systems.

The theoretical framework of complex neutrosophic graphs (CNGs) is a useful tool for illuminating and clarifying complicated and ambiguous difficulties faced in actual situations. This efficacy stems from its capacity to successfully communicate the intrinsic features of unpredictability, complexity, imprecision, and uncertainty connected with the things contained inside these sets. However, to address practical problems about truth membership, indeterminacy membership, and falsity membership functions, it is necessary to reformulate these techniques using precise numerical values. To address this issue, the idea of CTNGs was proposed, which employs linear t-norm and t-conorm operators. The implementation of CTNGs originates from the need for a systematic and adaptive technique to successfully manage ambiguity and enable decision-making based on pre-defined criteria. In this case, the parameter ‘t’ simplifies the procedure by defining specific criteria for determining the degree of truth membership, indeterminacy membership and falsity membership. In many real circumstances, decision-making must account for varied levels of confidence. Introducing the parameter ‘t’ in CTNGs seeks to overcome standard NFG limits by providing precise control over stringency, improving customization, allowing for independent decision thresholds, increasing flexibility, and decreasing ambiguity. These advantages establish CTNGs as an effective strategy for representing uncertainty and promoting well-informed decision-making in situations demanding a personalized and controlled approach to uncertainty management.

The benefit of CTNGs extends to comprehending and negotiating complicated choice settings, where standard NFG fails. These graphs offer decision-makers significant tools for studying and evaluating various options by completely illustrating the complex interplay between input and outcome factors. Complex fuzzy connections enable decision-makers to analyze numerous criteria and their interdependence methodically, allowing for a more holistic approach to tackling complex decision-making difficulties. The complex approach of CTNGs provides a considerable improvement in decision-making, especially in scenarios defined by truth membership, indeterminacy membership, falsity membership, and the parameter ’t’. It represents a break from the constraints imposed by binary logic, providing the path for greater precision in decision-making processes. Finally given some exiting work in Table 1.

Understanding complex interactions in many contexts, such as biodiversity conservation, is critical for making good decisions. However, standard approaches may fail to capture the complexities of such multiple events. As a result, there is a need for a sophisticated tool capable of correctly representing and analyzing complicated relationships.

The use of CTNGs as a unique tool for expressing and evaluating complex relationships is what distinguishes this work. While there are other graph-based models, including complex t-neutrosophic sets (CTNS) brings a new depth to the topic. This integration allows for the depiction of uncertainty, indeterminacy, and inconsistency in complex systems, making the suggested technique ideal for modeling real-world events with various interacting elements. Furthermore, the paper investigates basic set operations, homomorphism, and isomorphism within the context CTNGs, adding to its originality.

The main contribution of this study is the development of CTNGs, an effective tool for visualizing and understanding complex relationships, especially in the context of biodiversity protection. The work illustrates how CTNGs effectively capture complex interactions between several conservation variables, supporting well-informed decision-making. By means of real-world implementations, it demonstrates the adaptability of CTNGs in tackling issues related to biodiversity conservation by taking into account several pertinent aspects including species preservation, habitat preservation, and climate change adaption. The incorporation of indeterminacy, truth membership, and falsity membership functions into CTNGs gives decision-makers an understanding of conservation endeavors, empowering them to efficiently prioritize actions. Furthermore, the parameter ’t’ in CTNGs enables modification to fit various sensitivities and situations, enabling decision-makers to manage risk and uncertainty and navigate through complex decision-making environments. All things considered, this study provides a thorough framework and analytical tools that improve biodiversity conservation decision-making processes, greatly advancing ecosystem preservation and human well-being.

The paper progresses as follows: The ‘Basics of CTNGs’ section clarifies key terminology to highlight the originality of the offered work. Various set-theoretical operations are investigated with graphical representations in the next section, ‘Symmetric Operations on CTNGs’. Following that, the section ‘Isomorphism CTNGs’ defines homomorphisms and isomorphisms inside CTNGs. Moving on, the ‘Application of CTNG in biodiversity conservation’ section employs the newly developed strategy to promote ecosystem preservation. Finally, the research finishes with sensitivity analysis, comparative analysis, and definite conclusions summarizing the findings.