Abstract
Fuzzy topological indices are one of the accomplished mathematical approaches for numerous technology, engineering, and real-world problems such as telecommunications, social networking, traffic light controls, marine, neural networks, Internet routing, and wireless sensor network (Muneera et al. (2021)). This manuscript comprises the study of a particular class of graphs known as snake graphs. Some innovative results regarding fuzzy topological indices have been established. The major goal of the work is to introduce the notions of First Fuzzy Zagrab Index, Second Fuzzy Zagrab Index, Randic Fuzzy Zagrab Index, and Harmonic Fuzzy Zagrab Index of the snake graph.
1. Introduction
One cannot deny the importance of mathematics in all professions around the globe as every facet of life is vastly dependent on the use of arithmetic and numbers. Mathematics is used to develop and interpret its theories, especially chemistry, astronomy, geography, physics, etc. Learning of data entry for numerous purposes (such as computer programming, engineering, science, banking, and accounting) entirely depends upon mathematics.
Binary mathematics is the heart of computer operation which is the essential form of math used in computer science, statistics, algebra, and calculus. Creationof new generation tools and innovative and modern technological such as the Internet is one of the core necessities of modern life. Mathematics is commonly used in animation. This allows the animator to discover the unknown from a simple set of equations and extrapolate aspects of geometric figures when working with objects that move and change. The animation designer uses mathematical techniques to show how the drawings are animated and converted and also zoomed in and out. Information rising after cognition and computational perception, i.e., uncertain, partially true, vague, imprecise, or without limitations, can be resolved through fuzzy logic. Moreover, it leads to an effective resolution through better assessment and multiple criteria for options for the desired problem. Innovative computing techniques developed through fuzzy logic are used in the expansion of intelligence systems for identification, pattern recognition, control, and optimization.
Flow problems are one of the widespread applications of graph theory, which include real-life scenarios such as the scheduling of airlines or moving from one place toanother place with least possible time and by spending less money. Airlines have to fly around the world along with operating crew for each plane. Employees might belong to a particular city, so it is difficult for all the workers to be available for all flights. Graph theory is used in order to schedule the flight for crews. For this purpose, directed graphs are used where the flights are taken as the input, the destination is identified through an arrow (direction), and the cities are treated as vertices. The network flow can be seen in the resulting graph. Moreover, weights can be assigned to edges or vertices, showing the importance of a city or the need for flights to move from one city to another city so that passengers can be moved from one place to another place appropriately.
Travelling from one place to another place is a routine matter. Due to population growth and a large number of wheels on the roads, it is difficult to reach your destination within time during peak hours. Smart phones are frequently used nowadays not only for communication purposes but also they help us schedule the daily life activities, and GPRS helps us to save our time during peak hours, find the directions to our desired location, and reach our destination in the shortest possible time with the help of graph theory as the said problem lies under the category of shortest path problem, which is commonly used in graph theory. For this purpose, the current location of the user is the starting point/first vertex, and the destination is the end point/last vertex. The streets are treated as the edges and a directed graph is used for the directions, and the arrow guides the user where to go. The estimated time to reach the desired location varies with the speed of the wheels. Different colours are used to show the rush of traffic on the road, and directions can be relocated in case of any wrong turn or change of route for uncertain reasons.
It is pretty famous not only in the field of mathematics but in every field of science that graphs are simply models of relationships. A graph is a suitable way of representing information and connecting relationships between entities. The entities are characterized by vertices and relations by edges. When there is fuzziness in the description of the entities, their relationships, or both, it is natural to draw a “Fuzzy Graph.”
Expansion in the field of graph theory has given us very remarkable and innovative directions such as the fuzzy graph theory, hypergraph theory, spectral graph theory, and most importantly in the field of Computer Science. Graph algorithms are a major part of this field. Some popular graph algorithms are breadth-first search, depth-first search, Dijkstra’s algorithm, algorithm, shortest path algorithm, minimum spanning tree algorithm, and maximum flow algorithm. Behind every algorithm, there is a deep knowledge of python coding. Due to this reason, computer languages are playing a beautiful role in improvising the networking analysis. The use of Python language in graph theory makes it beautiful and more innovative due to the diversity of the language. These days AI (artificial intelligence) has a great boom in the field of computer science and also graphs are playing a vital role in this field particularly in machine learning. The most recent use of the graph theory with machine learning is seen in the FIFA football world cup. By using machine learning and graph theory, they will monitor 19 different body parts of a player. This will help the referee to conduct a smooth game. In the same context, fuzzy graph theory is also used in social networking sites such as Facebook, Twitter, Instagram, WhatsApp, and Research gate day-by-day. Fuzzy methods and fuzzy graph theory are employed in machine learning and data mining [1].
Rosenfeld introduced Fuzzy graphs a decade later Zadeh’s landmark research article “Fuzzy Sets”. He established some relations regarding the properties of path graph, trees, and various graphs. The concept of fuzzy cut nodes and fuzzy bridges was established by Bhattacharya in [2]. Many properties of fuzzy graphs are similar to those of crisp graphs as fuzzy graphs are the generalization of the crisp graph. A crisp graph is a one-to one correspondence from .
In dynamical systems where the values of all state constraints are known, the fuzzy graph theory is effectively applicable, which is helpful for spotting initial behaviors in consecutive changes of assemblages of organisms. Using fuzzy relations to describe succession from the same dataset, the compositional dynamics of the species was examined to get the desired result.
One cannot deny the importance of graph theory in numerous fields of life such as to elaborate chemical structures in Chemistry and Biochemistry; in Computer Science, several algorithms are used to solve various road networks problems to solve traffic problems, to connect numerous cities through trains and air lines effectively to move great number of passengers in least possible trips in rail and flight networks, to establish the finest timetable, and to have the same teaching load for all faculty in an organization. Labeling of a graph is a practice where symbols can be assigned to elements of the graph according to our condition; sometimes, it is not essential to allocate labels to all members of the graph. Labeling of a graph has massive applications in several fields of life and science. For further study on fuzzy algebraic points of view, see [3–7].
Bibi and Devi [8] develop a new idea of fuzzy magic labeling of cycle and star graphs. Vimala and Nagarani [9] gave the generalized concept of energy for fuzzy labeling graph. Nagoor Gani et al. [10] describe various properties of fuzzy labeling of path graph and star graph. Numerous definitions and basic concepts are discussed in [2, 11, 12]. Mufti et al. [13] computed first and second fuzzy Zagrab indices of linear and multiacyclic hydrocarbons. Also one can find more details in [14–16].
In 2023, Islam and Pal [17] also discussed the Second Zagreb Index and its use in mathematical chemistry. Islam and Pal [18] presented the First Zagreb Index for fuzzy graphs and discussed its application too. In [19], a new index, the F-index, has been already investigated and improved. In 2022, Rajeshwari et al. [20] established the fuzzy topological indices for bipolar fuzzy graphs.
Barrientos [21] introduced the concept of -snake or cyclic snake as a natural extension of triangular snake graphs already defined by Rosa [22]. A graph which is connected having isomorphic cycles of , such that the block-cut point graph is a path known as snake graph. This paper deals with the First Fuzzy Zagrab Index, Second Fuzzy Zagrab Index, Randic Fuzzy Zagrab Index, and Harmonic Fuzzy Zagrab Index of snake graph.
2. Motivation
For topological indices in crisp graphs, numerous results and applications are available. Some circumstances cannot be handled using crisp graphs in many real-life problems. Therefore, to handle such problems, new topological indices are needed to be introduced for the solution of real-world problems in the fields of medicine, engineering, technology, social sciences, telecommunication, and many more. The motivation of our paper was based on the paper [23], in which they introduce fuzzy topological indices, and we are motivated to work on the general structures of different graphs. Therefore, we have computed the fuzzy topological indices for the snake graph.
3. Preliminaries
In this section, we will define some basic definitions.
Definition 1. (see [23]). Let be a finite, undirected fuzzy graph (without loop and parallel edges). Let be a finite set of vertices and be a finite set of edges. Two mappings and , where and such that for every pair of .
Definition 2. (see [23]). Let be a fuzzy graph. The notion of the degree of a vertex in the fuzzy graph is defined as the sum of all the weights of edges corresponding to vertex , i.e.,
Definition 3. (see [23]). Let be a fuzzy graph. The order of a fuzzy graph is defined as the sum of all the weights of vertices, i.e.,
Definition 4. (see [23]). Let be a fuzzy graph. The size of a fuzzy graph is defined as the sum of all the weights of edges, i.e.,
Definition 5. (see [23]). Kalathian et al. [23] introduce the Fuzzy First Zagreb indices. Let G be the fuzzy graph with nonempty vertex set. The First Zagreb Index is denoted by and defined as First Fuzzy Zagreb Index.
Definition 6. (see [23]). The Second Zagreb Index is denoted by and defined as the Second Fuzzy Zagreb Index.
Definition 7. (see [23]). The Randic Index of fuzzy graph is defined as Randic Fuzzy Zagreb Index.
Definition 8. (see [23]). The Harmonic Index of G (H(G)) is defined as Harmonic Fuzzy Zagreb Index.
4. Limitations
Fuzzy Zagreb indices of snake graph and snake graph is applicable to undirected graphs as snake graph is an undirected graph.
5. Applications of Fuzzy Graphs
Fuzzy graphs are very important in real life. The application of fuzzy graphs falls in the different fields of life such as ecology, social networks, telecommunications, link predictions, identification of human trafficking, traffic light controls, marine, neural networks, Internet routing, and many more. In [24], the authors have discussed and deployed directed fuzzy graph to identify human trafficking. They investigated the traversal of humans from one place to another place and also elaborated their properties. Fuzzy graphs are also used in decision-making analysis as an application and in applied economics for management as an optimization problem [25–30].
6. Advantages of Fuzzy Graphs
Fuzzy graphs, which combine the concepts of fuzzy sets and graphs, offer several advantages in representing and analyzing complex systems that involve uncertainty and imprecision. Here are some advantages of fuzzy graphs:
6.1. Uncertainty Modeling
Fuzzy graphs provide a framework to represent and handle uncertainty in graph-based systems. They allow for the representation of imprecise or vague information that cannot be easily captured using traditional graphs.
6.2. Flexibility in Edge Weights
Fuzzy graphs allow for the assignment of fuzzy edge weights, which represent the degree of membership or possibility of a relationship between vertices. This flexibility enables a more nuanced representation of relationships compared to traditional graphs, where edge weights are usually binary or numeric.
6.3. Reasoning under Uncertainty
Fuzzy graphs provide a basis for reasoning and decision-making in situations where there is uncertainty. By applying fuzzy logic and fuzzy reasoning techniques to fuzzy graphs, it becomes possible to perform computations and make inferences in the presence of imprecise or incomplete information.
6.4. Robustness to Noise
Fuzzy graphs are less sensitive to noise or small variations in data compared to traditional graphs. The fuzziness introduced in the graph representation can help smooth out irregularities and provide a more robust analysis in the presence of noisy or incomplete data.
6.5. MultiCriteria Analysis
Fuzzy graphs are well-suited for multicriteria decision-making. By incorporating fuzzy sets and fuzzy logic, they allow for the consideration of multiple factors or criteria simultaneously, each with its own degree of importance or membership.
6.6. Human-Like Reasoning
Fuzzy graphs align with human reasoning processes, as humans often deal with imprecise and uncertain information. By using fuzzy graphs, it becomes possible to model and mimic human-like decision-making processes, enabling more human-compatible systems and algorithms.
6.7. Applications in Various Fields
Fuzzy graphs have found applications in diverse fields such as pattern recognition, image processing, expert systems, social networks, transportation systems, control systems, and optimization. Their ability to handle uncertainty makes them useful in situations where precise information is unavailable or difficult to obtain.
It is important to note that the advantages of fuzzy graphs depend on the specific problem domain and the quality of the fuzzy modeling and reasoning techniques employed. The effectiveness of fuzzy graphs may vary depending on the complexity of the system being modeled and the availability of accurate and relevant data.
7. Zagrab Indices of
A connected graph consisting of blocks, in which each one is isomorphic to cycle , such that the block-cut vertex is a path is termed as snake graph. A vertex in a graph is called a cut vertex if graph without that vertex has more components than the original graph. This graph was first introduced by Barrientos [21], as a generalization of the concept of triangular snake introduced by Rosa [22].
8. Our Main Results
Theorem 9. Let snake graph be a fuzzy graph, then the First Fuzzy Zagreb Index of snake graph is .
Proof. snake graph shown in Figure 1 consists of vertices and edges.
The representation of the vertex set is given as under: Weight of the all vertices is 0.6 and has a total count of where has weight 0.7 and other 2 have weights 0.3 and 0.4, respectively. Weight of the all vertices is 0.7 and has a total count of and all have a weight of 0.7. Weight of the all vertices is 0.8 and has a total count of and all have a weight of 0.9. Weight of the all vertices is 0.9 and has a total count of and all have a weight of 0.7.
Theorem 10. Let snake graph be a fuzzy graph, then the Second Fuzzy Zagreb index of snake graph is .
Proof. snake graph shown in Figure 1 consists of vertices and edges.
The representation of the edge set is given as under and also you can see in Figure 2: the edge set having weights of the vertices (0.6, 0.6) has three (03) types of representations. has a vertex degree (0.7, 0.7) and other 2 have (0.4, 0.7) and (0.7, 0.3), respectively. The edge set having weights of the vertices (0.6, 0.7) has two (02) types of representations. has a vertex degree (0.7, 0.7) and other has (0.4, 0.7). The edge set has weights of the vertices (0.7, 0.8) has one (01) type of representation. All have vertex degree (0.7, 0.9). The edge set has weights of the vertices (0.8, 0.9) has one (01) type of representation. All have a vertex degree (0.9, 0.7). The edge set having weights of the vertices (0.9, 0.6) has two (02) types of representations. has a vertex degree (0.7, 0.7) and other has (0.7, 0.3).
Theorem 11. Let snake graph be a fuzzy graph, then the Randic Fuzzy Zagreb Index of snake graph is .
Proof. snake graph shown in Figure 1 consists of vertices and edges.
The representation of the edge set is given as follows: the edge set having weights of the vertices (0.6, 0.6) has three (03) types of representations. has a vertex degree (0.7, 0.7) and other 2 have (0.4, 0.7) and (0.7, 0.3), respectively. The edge set having weights of the vertices (0.6, 0.7) has two (02) types of representations. has a vertex degree (0.7, 0.7) and other has (0.4, 0.7). The edge set having weights of the vertices (0.7, 0.8) has one (01) type of representation. All have vertex degree (0.7, 0.9). The edge set having weights of the vertices (0.8, 0.9) has one (01) type of representations. All have vertex degree (0.9, 0.7). The edge set having weights of the vertices (0.9, 0.6) has two (02) types of representations. has a vertex degree (0.7, 0.7) and other has (0.7, 0.3).
Theorem 12. Let snake graph be a fuzzy graph, then the Harmonic Fuzzy Zagreb Index of snake graph is .
Proof. snake graph shown in Figure 1 consists of vertices and edges.
The representation of the edge set is given as follows: the edge set having weights of the vertices (0.6,0.6) has three (03) types of representations. has a vertex degree (0.7, 0.7) and other 2 have (0.4, 0.7) and (0.7, 0.3), respectively. The edge set having weights of the vertices (0.6, 0.7) has two (02) types of representations. has a vertex degree (0.7, 0.7) and other has (0.4, 0.7). The edge set having weights of the vertices (0.7, 0.8) has one (01) type of representation. All have a vertex degree (0.7, 0.9). The edge set having weights of the vertices (0.8, 0.9) has one (01) type of representations. All have a vertex degree (0.9, 0.7). The edge set having weights of the vertices (0.9, 0.6) has two (02) types of representations. has a vertex degree (0.7, 0.7) and other has (0.7, 0.3).
9. Zagrab Indices of
Fuzzy Zagreb indices of snake graph has been discussed in this section (for odd .) Barrientos [21] introduced the concept of -snake or cyclic snake as a natural extension of triangular snake graphs already defined by Rosa [22]. A graph which is connected having isomorphic cycles of , such that the block-cut point graph is a path, is known as snake graph. This paper deals with the First Fuzzy Zagrab Index, Second Fuzzy Zagrab Index, Randic Fuzzy Zagrab Index, and Harmonic Fuzzy Zagrab Index of snake graph.
Theorem 13. Let snake graph be a fuzzy graph and is odd, then the First Fuzzy Zagreb Index of snake graph is
Proof. Let snake graph shown in Figure 3 consists of vertices and edges; if , and , then .
The representation of the vertex set can be considered if we consider a special case for odd . We consider and its vertex set representation is given as and , so implies . Weight of the all vertices is 0.08 and has a total count of where has weight 0.07 and other 2 have 0.03 and 0.11, respectively. Weight of the all vertices , , , , and is 0.09, , respectively, and all have a total count of .
Theorem 14. Let snake graph be a fuzzy graph and is odd, then the Second Fuzzy Zagreb Index of snake graph is for .
Proof. snake graph shown in Figure 3 consists of vertices and edges; if , and , then .
The representation of the vertex set can be considered if we consider a special case for odd . We consider and its vertex set representation is given as follows with the help of Table 1. Furthermore, and , so implies the edge set having weights of the vertices (0.08, 0.08) has three (03) types of representations. has a vertex degree (0.11, 0.11) and other 2 have (0.08, 0.11) and (0.11, 0.03), respectively. The edge set having weights of the vertices (0.08, 0.09) and having weights of the vertices (0.09,0.10) has a vertex degree having weights of the vertices (0.10, 0.11) have a vertex degree (0.07,0.09), having weights of the vertices (0.11, 0.12) have a vertex degree (0.09, 0.11), having weights of the vertices (0.12, 0.13) have a vertex degree (0.11, 0.13), and all have vertex count . The edge set has two (02) types of representations. has a vertex degree (0.13, 0.11) and other has (0.13, 0.08).
Theorem 15. Let snake graph be a fuzzy graph and is odd, then the Randic Fuzzy Zagreb Index of Snake graph is for .
Proof. snake graph shown in Figure 3 consists of vertices and edges; if , and , then .
The representation of the vertex set can be consider if we consider a special case for odd . We consider and its vertex set representation is given as follows with the help of Table 1. Furthermore, and , so implies the edge set having weights of the vertices (0.08, 0.08) has three (03) types of representations. has vertex degree (0.11, 0.11) and other 2 have (0.08, 0.11) and (0.11, 0.03), respectively. The edge set has weights of the vertices(0.08, 0.09) and has weights of the vertices (0.09, 0.10) has vertex degree has weights of the vertices (0.10, 0.11) and have a vertex degree (0.07, 0.09), has weights of the vertices (0.11, 0.12) have vertex degree (0.09, 0.11), has weights of the vertices (0.12, 0.13) have a vertex degree (0.11, 0.13), and all have vertex count . The edge set has two (02) types of representations. has a vertex degree (0.13, 0.11) and other has (0.13, 0.08).
Theorem 16. Let snake graph be a fuzzy graph and is odd, then the Harmonic Fuzzy Zagreb Index of snake graph is for .
Proof. snake graph shown in Figure 3 consists of vertices and edges; if , and , then .
The representation of the vertex set can be consider if we consider a special case for odd . We consider and its vertex set representation is given as follows with the help of Table 1. Furthermore, and , so implies the edge set having weights of the vertices (0.08, 0.08) has three (03) types of representations. has a vertex degree (0.11, 0.11) and other 2 have (0.08, 0.11) and (0.11, 0.03), respectively. The edge set has weights of the vertices (0.08, 0.09) and having weights of the vertices (0.09,0.10) has a vertex degree having weights of the vertices (0.10,0.11) have vertex degree (0.07, 0.09), having weights of the vertices (0.11, 0.12) have vertex degree (0.09, 0.11), has weights of the vertices (0.12, 0.13) and has a vertex degree (0.11, 0.13), and all have vertex count . The edge set has two (02) types of representations. has a vertex degree (0.13, 0.11) and other has (0.13, 0.08).
10. Graphical Analysis of
10.1. Conclusion
In this paper, First Fuzzy Zagreb Index, Second Fuzzy Zagreb Index, Harmonic, Randic Fuzzy Zagreb index, and Harmonic Fuzzy Zagreb Index of snake graph have been discussed and it has been observed that all of these indices can be calculated using formulas. We have computed the fuzzy topological indices for general structure, the comparison of snake graph with different values of and has also been discussed in Table 2, and graphical analysis is done in Figure 4. Using this approach, anyone can find the abovementioned indices of any general structure. This section is closed by raising the following open problem.
10.2. Open Problem
Fuzzy Zagreb indices of snake graph for even is still an open problem.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This study was supported by Researchers Supporting Project number (RSP2023R440), King Saud University, Riyadh, Saudi Arabia.