Strong Fuzzy Chromatic Polynomial (SFCP) of Fuzzy Graphs and Some Fuzzy Graph Structures with Applications

: In fuzzy graph theory, strong arcs have separate importance. Assign different colors to the end nodes of strong arcs in the fuzzy graph is strong coloring. Strong coloring plays an important role in solving real-life problems that involve networks. In this work


Introduction
Fuzzy graph theory is a dominant concept for modeling and solving combinatorial optimization problems which come from different fields. On the basis of Zadeh's fuzzy sets [1], the theory of fuzzy graph was narrated by Rosenfeld [2]. The author has studied the fuzzy analogs of several basic graph theoretical concepts. Bhutani [3] introduced the concept of automorphisms of fuzzy graphs and defined a complete fuzzy graph. Mordeson and Nair [4] have established the necessary and sufficient condition for a fuzzy graph which is cycle to be the fuzzy cycle. The fuzzy tree was characterized by Sunitha and Vijayakumar [5] using its unique maximum spanning tree. Strong arcs in a fuzzy graph were introduced by Bhutani and Rosenfeld [6]. The authors have also studied strong arcs in a fuzzy tree. Classifications of arcs in a fuzzy graph are very helpful to understand the entire structure of the fuzzy graph. Based on the strength of an arc, Mathew and Sunitha [7] have classified arcs in the fuzzy graph. In [8], the authors introduced the strongest strong cycle in fuzzy graphs. Lately, many researchers have actively worked on advancing fuzzy graphs [9][10][11][12][13]. Besides, they have studied fuzzy graph structures including strong, complete, regular fuzzy graph structures [14][15][16]. More recently, various types of fuzzy graphs such as bipolar fuzzy graphs [17], m-polar fuzzy graphs [18] Pythagorean fuzzy graphs [19], Dombi fuzzy graphs [20], Pythagorean dombi fuzzy graphs [21], Picture fuzzy graphs [22] have studied by different scholars. Moreover, several researchers published their work on applications of types of fuzzy graphs [18,[23][24][25][26] and index of fuzzy graphs [27,28].
In fuzzy graph theory, coloring plays a paramount role in solving optimization problems. Munoz et al. [29] introduced the idea of coloring fuzzy graphs. After that, Eslahchi and Onagh [30] have defined the fuzzy chromatic number of a fuzzy graph. Several authors including Kishore and Sunitha [31] and Samanta et al. [32], Mahapatra et al. [33] have worked on fuzzy coloring. In [34], Kishore and Sunitha have initiated the concept of the strong coloring of fuzzy graphs based on strong arcs. The authors defined the strong chromatic number of a fuzzy graph. Recently, Rosyida et al. [35] introduced a fuzzy chromatic number in the union of fuzzy graphs. Mamo and Srinivasa Rao [36] introduced the concept of the fuzzy chromatic polynomial of a fuzzy graph based on -cuts.
The concept of strong coloring has a significant role in addressing real-life problems that involve networks. The core contributions of this research article are as follows.
i. As far as we know, there exists no research work on SFCP until now. Hence, in this article, we present the definition of SFCP of a fuzzy graph based on strong coloring. ii. We establish the necessary and sufficient condition for SFCP of a fuzzy graph and chromatic polynomial of its underlying crisp graph is equivalent. iii. We study the SFCP of some structures of fuzzy graphs such as strong fuzzy graphs, complete fuzzy graphs, fuzzy cycles, and fuzzy trees. iv. We obtain relations between SFCP and fuzzy chromatic polynomial of strong fuzzy graphs, complete fuzzy graphs, and fuzzy cycles. v. We present the definition of a strong fuzzy chromatic number of a fuzzy graph in terms of SFCP. vi. Dual applications of SFCP in traffic flow problems are described in this article. Also, we suggest SFCP approach to solve strong coloring problems.

Preliminaries
In this section, we reviewed some basic definitions and concepts on fuzzy graphs, strong arcs and strong coloring of fuzzy graphs, which are important for present work. The following basic definitions and related concepts are taken from [3,7,34,[36][37][38].
Definition 1. A fuzzy graph = ( , , ) is a triple consisting of a nonempty set V together with a pair of functions : → [0, 1] and : → [0, 1] such that for all , ∈ , ( , ) ≤ ( ) ∧ ( ). Here, the fuzzy set is called the fuzzy vertex set of G and the fuzzy edge set of G.
Definition 11. If 8 = ; , and @ ≥ 3 then is called a cycle and is called a fuzzy cycle, if it contains more than one weakest arc.
Definition 12. The strength of is defined to be B ( <=9 , < ) ; <C9 . In words, the strength of a path is defined as the degree of membership of the weakest arc in . We denote the strength of a path P by D( ).
Definition 13. The strength of connectedness between two vertices and ! is defined as the maximum of the strengths of all paths between and !and is denoted by EFGG H ( , !) or 7 ( , !). The strongest path joining any two vertices , ! has strength 7 ( , !). and ! in a fuzzy graph obtained from by deleting the arc ( , !).
Here, a strong arc is either -strong or K-strong. Also, a Larc ( , !) is called a L * -arc if ( , !) > ( , ) where ( , ) is the weakest arc of . Definition 18. Consider a fuzzy graph = ( , , ). The coloring E ∶ ( ) → ℕ (where ℕ is the set of all positive integers) such that E( ) ≠ E(!) if ( , !) is a strong arc (strong and K-strong) in is called strong coloring.
Definition 19. A fuzzy graph is k-strong colorable if there exists a strong coloring of from a set of P colors. Definition 20. The minimum number P for which is Pstrong colorable is called the strong chromatic number of denoted by Q R ( ). Definition 21. For a fuzzy graph , the fuzzy chromatic polynomial of is denoted by -S , P and defined as the chromatic polynomial of its crisp graphs -, for ,.

Strong Fuzzy Chromatic Polynomial of a Fuzzy Graph
In the sense of arcs, chromatic polynomials of crisp graphs are always strong chromatic polynomials. Since in crisp graph theory, all the arcs are strong by nature. But, in fuzzy graph theory, arcs are different and have separate significance. In this section, we introduce the new concept, strong chromatic polynomial in a fuzzy graph, called strong fuzzy chromatic polynomial (SFCP). Also, we define a strong fuzzy chromatic polynomial of a fuzzy graph based on strong coloring.
A strong fuzzy chromatic polynomial counts the number of strong coloring on the vertices of a fuzzy graph and is defined as follows.
Definition 22. Let be a fuzzy graph with a positive integer P, the number of distinct P-strong colorings of is called strong fuzzy chromatic polynomial (SFCP) of . It is denoted by R S , P . Example 1. Consider the fuzzy graph given in Figure 1. In Figure 1, we found that arc , ! is -strong since Here, strong coloring gives P color for and P Z 1 color for both ! and V (See Figure 2). Hence, the SFCP of is P P Z 1 P Z 1 . That is, Theorem 2 shows us how to determine the strong fuzzy chromatic polynomial of a fuzzy graph in which some of the arcs are not strong. Theorem 2. Let be a fuzzy graph in which some of its arcs are not strong. Then R S , P , P , where be a fuzzy subgraph obtained from by deleting L-arcs and be its underlying crisp graph.
Proof. Let be a fuzzy graph. Suppose is a fuzzy subgraph obtained from by deleting L -narcs.
Thus, contains only strong arcs. Therefore, by Theorem 1 we have, Since is a fuzzy subgraph of , all the arcs of are the only strong arcs of . This implies that Therefore, from equation (1) and (2), we get, R S , P , P

Strong Fuzzy Chromatic Polynomial of Some Fuzzy Graph Structures
In the section, we study the SFCP of some fuzzy graph structures and their relations with the fuzzy chromatic polynomial.

Strong Fuzzy Graphs
In this subsection, we study the SFCP of strong fuzzy graphs and the relation between SFCP and fuzzy chromatic polynomial of a strong fuzzy graph. Theorem 3. Let is a strong fuzzy graph and * be its underlying crisp graph. Then R S ( , P , P .
Proof. Suppose is a strong fuzzy graph and is its underlying crisp graph. Since is a strong fuzzy graph, then by Theorem 4.1 of [36], all the arcs in are strong. Then, by Theorem 1, the result holds. Remark 1. The Converse of Theorem 3 does not hold generally.
Example 2. Let us consider the fuzzy graph , shown in Figure 3. Remark 2. If all the arcs in a fuzzy graph are strong, then need not be a strong fuzzy graph (See Figure 3). The relation between SFCP and fuzzy chromatic polynomial of a strong fuzzy graph is established below.
Theorem 4. Let G be a strong fuzzy graph, then ^ " Proof. Let be a strong fuzzy graph. Let be the underlying crisp graph of . Let " be a level set of . If we take " and min " and since is a fuzzy graph.
Then by Theorem 40 of [36], -S , P , P Since is a strong fuzzy graph, then by Theorem 2, we have R S , P , P Therefore, from equation (3) and (4), the result holds immediately.

Complete Fuzzy Graphs
In this subsection, the SFCP of complete fuzzy graphs and the relation between SFCP and fuzzy chromatic polynomial of a complete fuzzy graph are studied.  Figure 4. In , the arcs , and !, V are -strong and the remaining arcs are Kstrong. So, all the arcs in are strong. Therefore, by Theorem 1, we have R S , P , P which is equal to a b , P . But, is not a complete fuzzy graph. Remark 4. If all the arcs in a fuzzy graph are strong, is not necessarily a complete fuzzy graph. (See Figure 4) Remark 5. A complete fuzzy graph is strong whereas a strong fuzzy graph need not be complete. Remark 6. If a fuzzy graph G is strong or/and complete fuzzy graph, then R S , P , P .
The relation between SFCP and fuzzy chromatic polynomial of a complete fuzzy graph is established in Theorem 6. Theorem 6. If is a complete fuzzy graph, then there exist , such that R S , P -S , P .
Proof. Suppose is a complete fuzzy graph with n vertices. Define , " + 0 , where " is the level set of . If 0 and since is a fuzzy graph, then by Theorem 39 of [36].
-S , P a ; , P Since is a complete fuzzy graph, by Theorem 5 we have R S , P a ; , P Therefore, from equation (6) and (7), we get,

Fuzzy Cycles
In this subsection, the SFCP of a fuzzy cycle and the relation between SFCP and fuzzy chromatic polynomial of a fuzzy cycle are discussed.
In fuzzy cycle , there are no L-arcs. In other words, the weakest arcs in G are K-strong and all the remaining arcs are -strong [6]. Theorem 7. Let be a fuzzy graph such that is a cycle. Then is a fuzzy cycle if and only if R S , P , P .
Proof. Let be a fuzzy graph such that is a cycle. Suppose is a fuzzy cycle. Since is a fuzzy cycle, then contains only strong arcs [7]. Therefore, by Theorem 1, the result immediately holds.
On the other hand, we shall prove by contrapositive of if Proof. The result immediately holds from Theorem 7 and Remark 52 of [36].
The relation between SFCP and fuzzy chromatic polynomial of a fuzzy cycle is established below. Theorem 8. Let be a fuzzy graph and be a cycle. If is a fuzzy cycle, then there exist " such that R S , P -S , P .
Proof. Let be a fuzzy graph such that is a cycle. Suppose is a fuzzy cycle. Since is a fuzzy cycle. Then by Theorem 7, we get R S , P , P Now take min " , where L is the fundamental set of , by Theorem 40 of [36], we have -S , P , P From equations (8) and (9), we get R S , P -S , P .
Therefore, for a fuzzy cycle , there exists " such that SFCP of a fuzzy cycle is equal to the fuzzy chromatic polynomial of a fuzzy cycle.

Fuzzy Trees
In this subsection, we discuss the SFCP of a fuzzy tree and a fuzzy graph whose is a tree and not the tree. Theorem 9. Let be a fuzzy graph such that is a tree. Then R S , P , P .
Proof. Let be a fuzzy graph such that is a tree. is a tree by Definition 9. Since is a tree, clearly, all the arcs of are fuzzy bridges. Then all the arcs in are strong by Proposition 2 of [6] and Theorem 4 of [5]. Therefore, the result holds by Theorem 1.
Remark 7. For fuzzy graph with n vertices such that is a tree, then h ; , where h ; is a tree with n vertices.
Moreover, by Theorem 9, we have R S , P h ; , P .
Theorem 10 gives the SFCP of a fuzzy tree is equal to the SFCP of its maximum spanning tree.
Theorem 10. Let be a fuzzy graph such that is not a tree. If is a fuzzy tree, then R S , P R S h, P , where h is the maximum spanning the tree of .
Proof. Let be a fuzzy graph and be not a tree. Suppose is a fuzzy tree. Since is a fuzzy tree, then has a unique maximum spanning tree T by Theorem 10 of [5].
Since is a fuzzy tree and is not a tree, by Theorem 5 of [5] consists of at least one L-arc. Then, by Proposition 4 of [6], the strong arcs of are arcs of h of . This implies that all the arcs of h are strong.
Hence, the strong coloring of is exactly the same as the strong coloring of hof . Hence, R S , P R S h, P .
Example 4. Consider the following fuzzy tree in Figure  5 and its maximum spanning tree h in Figure 6.  In Figure 5, arc , ! and arc , V are -strong but !, V is L -arc, whereas, in Figure 6, both arcs , ! and , V are -strong which are strong arcs of a fuzzy tree in Figure 5. In and h, vertex can be strong colorable in P ways and vertices ! and V can be strong colorable in P Z 1 ways. Therefore, R S ( , P P P Z Where h is the underlying crisp graph of T.
Since is a fuzzy tree, then by Theorem 10, we have Hence, from equation (10) and (11) Example 5. Consider the fuzzy tree in Figure 7 (a) and its maximum spanning tree T of in Figure 7  Theorem 12. Let be a fuzzy graph such that is a cycle. If is not fuzzy tree then R S , P , P Proof. Let be a fuzzy graph and be a cycle. Suppose is not a fuzzy tree. Then, by Theorem 1 of [4]. is a fuzzy cycle. Since is a fuzzy cycle, then the result immediately holds by Theorem 7.
Besides counting the strong colorings on fuzzy graphs, SFCP can be used to get the strong fuzzy chromatic number of a fuzzy graph.
The following definition tells us how to use the SFCP to find the least number of colors for which a fuzzy graph to be strong colorable. i.e. P Q R S . Note that strong fuzzy chromatic polynomials are powerful mathematical tools. Once SFCP of a fuzzy graph is determined, it is a simple and shortcut method for solving strong coloring problems that are modeled by fuzzy graphs.

Applications of Strong Fuzzy Chromatic Polynomial
In this section, we discuss two applications of the strong fuzzy chromatic polynomial in vehicular traffic flow problems at road intersections. In the first application, we apply our proposed SFCP to find a minimum number of traffic light phases, in the second application to compute the possible number of traffic light patterns with the optimized phases.

Application for Finding the Minimum Number of Traffic Light Phases
In traffic flow problems, phases collect non-conflict movements of traffic flows at road intersections so that the traffic flows being safe and smooth. Phase design is the first step in the traffic signal designing procedure. This shows that phase designing is a very important step since the further steps are affected by it. Besides, a minimum number of phases are very helpful in reducing the waiting time of vehicles and fuel consumption. In this subsection, we apply the proposed SFCP to find the minimum number of traffic light phases at a road intersection. Now, we demonstrate the first application of SFCP theory to traffic flow problems with the help of Example 6. Example 6. Consider the traffic flow problem of a city in Figure 8. In this problem, we consider a four-legged intersection through four traffic flows. Each flow can move straight, turn right and turn left at the road intersection. We assume that the number of vehicles at each lane is not equal. Suppose that the traffic volumes of flow 1, flow 2, flow 3 and Fuzzy Graphs and Some Fuzzy Graph Structures with Applications flow 4 are High (0.9), Low (0.2), Medium (0.5) and Low (0.2), respectively. The fuzzy graph model for the traffic flow problem in Figure 8 base on Case 1 is shown in Figure 9. Case 2: When >, l p \>@ > , l for > and l with low traffic (where "p" represents much less than).
Suppose 2, 4 q p 0.2. So, the fuzzy graph model 9 for the traffic flow problem in Figure 8 based on Case 2, is shown in Figure 10. Next, identify the types of arcs in the fuzzy graph models and 9 , since the fuzzy graph model shown in Figure 9 is a complete fuzzy graph, by Proposition 3.13 of [3] and Proposition 2 of [6], all arcs of a complete fuzzy graph are strong.
On the other hand, by routine computation, for the fuzzy graph model 9 shown in Figure 10, we found that arc (1,3) is -strong, arcs (1,2), (1,4), (2,3) and (3,4) are K-strong and arc (2,4) is L-arc. Hence, arcs (1,2), (1,3), (1,4), (2,3) and (3,4) are the only strong arcs in 9 . Now, determine the SFCP for the fuzzy graph model and 9 based on their arcs. Since is a complete fuzzy graph, by Theorem 5, R S , P a b , P , where a b is the complete crisp graph. But, a b , P P P Z 1 P Z 2 P Z 3 Therefore, R S , P P P Z 1 P Z 2 P Z 3 Analogously, since all the arcs in 9 are not strong (b/c arc (2, 4) is L-arc). Suppose is a fuzzy subgraph obtained from the fuzzy graph 9 by deleting L -arc. Explicitly, 9 Z 2, 4 is given in Figure 11. Therefore, by Theorem 2, R S 9 , P , P where is the underlying crisp graph of is shown in Figure 12.  By deletion contraction algorithm, we obtain,

Result and Discussion
Consider the SFCP of in (14), and using Definition 23, we get, Q R S 4.
This result shows that there is a minimum of 4 traffic light phases are needed to control the traffic flow in Figure 8. With these 4 phases, the traffic flows at the intersection being smooth and safe.
Analogously, consider the SFCP of 9 in (15), and using Definition 23, we obtain, Q R S 9

3.
This result shows that there is a minimum of 3 traffic light phases are needed to control the traffic flow in Figure 8. For instance, in phase 1, flow 1 can proceed, in phase 2, flow 3 can proceed and in phase 3, both flows 2 and 4 can go simultaneously at the intersection without making any conflict, however, the flows 2 and 4 have a conflicting movement originally.
In model , the possibility of an accident occurs between each pair of conflict flows that are moving at the intersection is significant since all the flows are grouped into independent phases. But, in model 9 , the possibility of accident occurs between conflict flows 2 and 4 are insignificant (i.e.
2,4 q since they are grouped in the same phase.
Therefore, the fuzzy graph model is very helpful and flexible for describing and handling the uncertain information in traffic flow problems and SFCP is a powerful mathematical tool for optimizing the traffic light phases at road intersections.

Application for Obtaining the Possible Number of Traffic Light Patterns
Most of the time, the traffic light pattern which we are going to plan at road intersection might not be appropriate, efficient and reduce the vehicle waiting time as compared with each other. Therefore, choosing an efficient and appropriate traffic light flow pattern with less waiting time is a major issue in traffic light control and management.
In this subsection, we apply SFCP to obtain the possible number of traffic light flow patterns with the optimized phases.
In Example 6, for the model in Figure 9, we have R S , P P P Z 1 P Z 2 P Z 3 And Q R S 4.
Therefore, R S r , Q R S s 24.
Here, the result shows that there are 24 possible traffic light flow patterns with 4 phases.
Similarly, for the model 9 in Figure 10 , we have R S 9 , P P P Z 1 P Z 2 [ And Q R S 9

3.
Therefore, we obtain R S r 9 , Q R S 9 s 6.
Here, the result shows that there are 6 possible traffic light flow patterns with 3 phases.
Note that all of these traffic light flow patterns are nonconflicting combinations. The six possible traffic light flow patterns with 3 phases are depicted in Tables 1-6.     Computing the SFCP of the fuzzy graph at the minimum number of phases gives the possible number of traffic light patterns for the given traffic flow problem. Sometimes all the possible patterns which we have obtained are not equally appropriate and efficient. If this is the case, the SFCP is very helpful to choose the most appropriate and efficient patterns with reduced vehicle waiting times.

Conclusion
In this paper, we have introduced the concept of the strong fuzzy chromatic polynomial of fuzzy graphs and we have defined SFCP of fuzzy graph based on the strong coloring of a fuzzy graph. We have established the necessary and sufficient condition for SFCP of the fuzzy graph to be the chromatic polynomial of its underlying crisp graph. Furthermore, we have studied the SFCP of strong fuzzy graphs, complete fuzzy graphs, fuzzy cycles, and fuzzy trees. Also, we have given relations between SFCP and fuzzy chromatic polynomial of strong fuzzy graphs, complete fuzzy graphs, and fuzzy cycles. Finally, applications of SFCP theory to traffic flow problems are offered to demonstrate the applicability of the proposed work. In the future, we will develop a SFCP algorithm to solve strong coloring problems.