Fuzzy Chromatic Polynomial of Fuzzy Graphs with Crisp and Fuzzy Vertices Using α-Cuts

Coloring of fuzzy graphs has many real life applications in combinatorial optimization problems like traffic light system, exam scheduling, register allocation, etc. In this paper, the concept of fuzzy chromatic polynomial of fuzzy graph is introduced and defined based on α-cuts of fuzzy graph. Two different types of fuzziness to fuzzy graph are considered in the paper. The first type was fuzzy graph with crisp vertex set and fuzzy edge set and the second type was fuzzy graph with fuzzy vertex set and fuzzy edge set. Depending on this, the fuzzy chromatic polynomials for some fuzzy graphs are discussed. Some interesting remarks on fuzzy chromatic polynomial of fuzzy graphs have been derived. Further, some results related to the concept are proved. Lastly, fuzzy chromatic polynomials for complete fuzzy graphs and fuzzy cycles are studied and some results are obtained.


Introduction
Nowadays, many real world problems cannot be properly modeled by a crisp graph theory, since the problems contain uncertain information. The fuzzy set theory, anticipated by Zadeh [1], is used to handle the phenomena of uncertainty in real life situation. A lot of works have been done in fuzzy shortest path problems using type 1 fuzzy set in [2][3][4][5]. Dey et al. [6] introduced interval type 2 fuzzy set in the fuzzy shortest path problems. Recently, in [7], the authors proposed a genetic algorithm for solving fuzzy shortest path problem with interval type 2 fuzzy arc lengths. Some researchers also used the fuzzy set theory to touch the uncertainty in crisp graphs. Kaufmann [8] proposed the first definition of fuzzy graph in 1973, based on Zadeh's fuzzy relations. Later, Rosenfeld [9] introduced another elaborated definition of fuzzy graph with fuzzy vertex set and fuzzy edge set in 1975. He developed the theory of fuzzy graph. After that, Bhattacharya [10] has established some connectivity concepts regarding fuzzy cut nodes and fuzzy bridges. Bhutani [11] has studied automorphisms on fuzzy graphs and certain properties of complete fuzzy graphs. Also, Mordeson and Nair [12] introduced cycles and cocycles of fuzzy graphs. Several authors including Sunitha and Vijayakumar [13,14], Bhutani and Rosenfeld [15], Mathew and Sunitha [16], Akram [17], and Akram and Dudek [18] have introduced numerous concepts in fuzzy graphs. Fuzzy graph theory has several applications in various fields like clustering analysis, database theory, network analysis, information theory, etc. [19].
Coloring of fuzzy graphs plays a vital role in theory and practical applications. It is mainly studied in combinatorial optimization problems like traffic light control, exam scheduling, register allocation, etc. [20]. Fuzzy coloring of a fuzzy graph was defined by authors Eslahchi and Onagh in 2004 and later developed by them as fuzzy vertex coloring [21] in 2006. Another approach of coloring of fuzzy graphs was introduced by Munoz et al. [22]. The authors have defined the chromatic number of a fuzzy graph. Incorporating the above two approaches of coloring of fuzzy graph, Kishore and Sunitha [23] introduced chromatic number of fuzzy graphs and developed algorithm. Dey and Pal [24] introduced the vertex coloring of a fuzzy graph using -cuts. In [25], they have used the vertex coloring of fuzzy graph to classify the accidental zone of a traffic control. Further, in [26], the authors proposed genetic algorithm to find the robust solutions for fuzzy robust coloring problem. The authors, Ananthanarayanan and Lavanya [20], introduced fuzzy chromatic number and fuzzy total chromatic number 2 Advances in Fuzzy Systems of a fuzzy graph using -cuts. Rosyida et al. [27] proposed a new approach to determine fuzzy chromatic number of fuzzy graph through its -chromatic number. Samanta et al. [28] also introduced a new concept for coloring of fuzzy graphs using fuzzy color. Lately, Dey et al. [29] introduced the concept of vertex and edge coloring of vague graphs which are the generalized structure of fuzzy graphs.
In the literature, to the best of our knowledge, there is no study on the fuzzy chromatic polynomial of fuzzy graphs. Therefore, in this paper, we consider the chromatic polynomial in fuzzy graph, called fuzzy chromatic polynomial of fuzzy graph. Based on this, we define the concept of fuzzy chromatic polynomial of a fuzzy graph usingcuts of fuzzy graph. Also, we determine the fuzzy chromatic polynomial for fuzzy graphs with crisp and fuzzy vertex set.
To determine the fuzzy chromatic polynomial, the classical method for computing the chromatic polynomial of crisp graph is used. Next, we derive some interesting remarks on fuzzy chromatic polynomial of fuzzy graph with crisp and fuzzy vertices. Further, we prove more elegant results on fuzzy chromatic polynomial of fuzzy graphs. Finally, we study a fuzzy chromatic polynomial for complete fuzzy graphs and fuzzy cycles.
The rest of paper is organized as follows. In Section 2, some basic definitions and elementary concepts of fuzzy set, fuzzy graph, and coloring of fuzzy graphs are reviewed. In Section 3, fuzzy chromatic polynomial of a fuzzy graph using -cut of fuzzy graph is defined. Also, fuzzy chromatic polynomials for fuzzy graphs with crisp and fuzzy vertices are determined. In Section 4, more results on fuzzy chromatic polynomials are proved. Furthermore, fuzzy chromatic polynomial for complete fuzzy graphs and fuzzy cycles are studied. Finally, the paper is concluded in Section 5.

Preliminaries
In this section, some basic aspects that are necessary for this paper are included. These preliminaries are given in three subsections.

. . Basic Definitions and Concepts on Vertex Coloring and
Chromatic Polynomial. In this subsection some basic definitions and concepts of vertex coloring and chromatic polynomials are reviewed [30,31].
Definition . Let G = (V, E) be a graph. A vertex-coloring of G is an assignment of a color to each of the vertices of G in such a way that adjacent vertices are assigned different colors. If the colors are chosen from a set of colors, then the vertex-coloring is called a -vertex-coloring, abbreviated to -coloring, whether or not all colors are used.
Definition . If G has a -coloring, then is said to becolorable.
Definition . The smallest , such that G is -colorable, is called the chromatic number of G, denoted by (G).
Definition . Let G be a simple graph. The chromatic polynomial of G is the number of ways we can achieve a proper coloring on the vertices of G with the given colors and it is denoted by (G, ). It is a monic polynomial in with integer coefficients, whose degree is the number of vertices of G.
If we are given an arbitrary simple graph, it is usually difficult to obtain its chromatic polynomial by examining the structure of a graph (by inspection). The following theorem gives us a systematic method for obtaining the chromatic polynomial of a simple graph in terms of the chromatic polynomial of null graphs.
Theorem 5. Let G be a simple graph, and let G − e and G/e be the graphs obtained from G by deleting and contracting an edge e. en . . Basic Definitions on Fuzzy Set and Fuzzy Graphs. In this subsection, some basic definitions on fuzzy set and fuzzy graphs are reviewed [1,9,19,32].

Definition .
A fuzzy set defined on a nonempty set is the family = {( , ( )) | ∈ }, where : → is the membership function. In classical fuzzy set theory the set I is usually defined as the interval [0, 1] such that It takes any intermediate value between 0 and 1 represents the degree in which ∈ . The set could be discrete set of the form = {0, 1, . . . , } where ( ) < ( ) indicates that the degree of membership of x to A is lower than the degree of membership of .
Definition . -cut set of fuzzy set is defined as is made up of members whose membership is not less than , In this paper, we denote ∧ V = min{ , V} and ∨ V = max{ , V}. Here, we considered fuzzy graph is simple (with no loops and parallel edges), finite, and undirected. is reflective (that is, (V, V) = (V), for all V ∈ ) and symmetric (that is, ( , V) = (V, ), for all ( , V) ∈ ).
Note that a fuzzy graph is a generalization of crisp graph in which (V) = 1 for all V ∈ and ( , V) = 1 if ( , V) ∈ and ( , V) = 0 if ( , V) ∉ . So, all the crisp graphs are fuzzy graphs but all fuzzy graphs are not crisp graphs.

Definition . For any fuzzy graph
. . , } and the set are called the fundamental sequence and the fundamental set (or level set) of , respectively.

. . Basic Definitions in Fuzzy Coloring of Fuzzy Graphs.
The concept of chromatic number of fuzzy graph was introduced by Munoz et al. [22]. The authors considered fuzzy graphs with crisp vertex set, that is, fuzzy graphs, for which (V) = 1 for all V ∈ and edges with membership degree in [0, 1].
Definition (see [21]). A family Γ = { 1 , 2 , . . . , } of fuzzy sets on a set is called a -fuzzy coloring of = ( , , ) if Definition (see [21]). The minimum number for which there exists a -fuzzy coloring is called the fuzzy chromatic number of , denoted as ( ).
Incorporating the features of the above two definitions, Kishore and Sunitha [23] modified the chromatic number ( ) of fuzzy graph as follows.

Fuzzy Chromatic Polynomial of a Fuzzy Graph
A fuzzy chromatic polynomial is a polynomial which is associated with the fuzzy coloring of fuzzy graphs. Therefore, chromatic polynomial in fuzzy graph is called fuzzy chromatic polynomial of fuzzy graph. In this section, we define the concept of fuzzy chromatic polynomial of a fuzzy graph based on -cuts of fuzzy graph which are crisp graphs. Furthermore, we determine the fuzzy chromatic polynomials for some fuzzy graphs with crisp and fuzzy vertices.
Crisp vertex coloring and the chromatic polynomial ofcut graph of the fuzzy graph are defined as follows.

Definition . Let
= ( , , ) be a fuzzy graph and denote -cut graph of the fuzzy graph which is a crisp graph Definition . The number of distinct k-coloring on the vertices of is called the chromatic polynomial of . It is denoted by ( , ).
Let { | ∈ } be the family of -cuts sets of , where the -cut of a fuzzy graph is the crisp graph . Hence, any crisp k-coloring can be defined on the vertex set of . The k-coloring function of the fuzzy graph is defined through this sequence. For each ∈ , let the fuzzy chromatic polynomial of be defined through a monotone family of sets.
Fuzzy chromatic polynomial of a fuzzy graph is defined as follows.
Definition . Let be a fuzzy graph. The fuzzy chromatic polynomial of is defined as the chromatic polynomial of its crisp graphs , for ∈ . It is denoted by ( , ).

. . Fuzzy Chromatic Polynomial of Fuzzy Graph with Crisp
Vertices. In this subsection, we present fuzzy chromatic polynomial of fuzzy graphs with crisp vertices and fuzzy edges.
A fuzzy graph with crisp vertices and fuzzy edges, and -cut graph of are defined as follows.

Advances in Fuzzy Systems
Definition . A fuzzy graph is defined as a pair = ( , ) such that (1) V is the crisp set of vertices (that is, ( ) = 1, ∀ ∈ ); (2) the function : x Definition . Let = ( , ) be a fuzzy graph. For ∈ ,cut graph of the fuzzy graph is defined as the crisp graph Example . Consider the fuzzy graph with crisp vertices and fuzzy edges in Figure 1.
In , we consider = {0, 0.2, 0.3, 0.5, 0.8, 1}; for each ∈ , we have a crisp graph and its chromatic polynomial which is the fuzzy chromatic polynomial of the fuzzy graph is obtained (see Figure 2). (The integers in the brackets denote the number of ways of coloring the vertices.) Remark . The fuzzy chromatic polynomial depends on the values of , which means the fuzzy chromatic polynomial varies for the same fuzzy graph for different values of .
For the fuzzy graph in Example 23, the fuzzy chromatic polynomial varies for different values of as shown below: Remark . The fuzzy chromatic polynomial of a fuzzy graph does not need to be decreased when the value of increases. Let us consider 1 = 0.2, 2 = 0.3, 3 = 0.5, 4 = 0.8, and 5 = 1 for the fuzzy graph in Example 23, but ( , ), ≥ , where is the chromatic number of the crisp graph i=1,2,3,4,5 is not decreased as { } increases. For instance, Remark . Let be a fuzzy graph with n vertices. Then for with crisp vertices and fuzzy edges and for = 1, ( , ) is not always ( , ), where is the null crisp graph with n vertices which is clear from the following example.
Example . Consider the fuzzy graph given in Figure 3(a). In Figure 3(a), for = 1, is a crisp graph in Figure 3(b). But is not a null crisp graph with three vertices. Thus, ( , ) ̸ = ( 3 , ).
The following result gives the degree of fuzzy chromatic polynomial of fuzzy graph with crisp vertices and fuzzy edges are equal to the number of vertices in crisp vertex set. Proof. Let = ( , ) be a fuzzy graph with crisp vertices and fuzzy edges. Now, We know that the degree of the chromatic polynomial of a crisp graph is the number of vertices of . That is, the degree of ( , ) = | |. By Definition 20, we have ( , ) = ( , ), for ∈ . From this it follows that the degree of ( , ) is equal to the degree of ( , ). Therefore, ( , ) = | |.

. . Fuzzy Chromatic Polynomial of Fuzzy Graph with Fuzzy
Vertices. In Section 3.1, we introduced the fuzzy chromatic polynomial of fuzzy graph = ( , ), where is crisp set vertices and is fuzzy set of edges. In this subsection, we present fuzzy chromatic polynomial of fuzzy graphs with fuzzy vertex set and fuzzy edge set. Here, we use Definition 8 for fuzzy graph and Definition 13 for -cut graph which are presented in Section 2.
Example . Consider a fuzzy graph with fuzzy vertex set and fuzzy edge set in Figure 4 In , we consider = {0, 0.2, 0.3, 0.4, 0.6, 0.8}; for each ∈ , we have a crisp graph and its chromatic polynomial which is the fuzzy chromatic polynomial of the fuzzy graph is obtained (see Figure 5). (The integers in the brackets denote the number of ways of coloring the vertices.) Remark . The fuzzy chromatic polynomial varies for the same fuzzy graph for different values of . It is clear from Example 29. Here, the fuzzy chromatic polynomial of the fuzzy graph in Example 29 is given below.
Advances in Fuzzy Systems  Remark . For fuzzy graph with fuzzy vertex set and fuzzy edge set, the degree of the fuzzy chromatic polynomial of is not always equal to the number of vertices of . It can be easily seen from the fuzzy chromatic polynomial of the fuzzy graph in Example 29. For instance, for = 0.4, the degree of ( , ) = 5 which is the number of vertices of , but for = 0.6, the degree of ( , ) = 3 ̸ = 5; also for = 0.8, the degree of ( , ) = 1 ̸ = 5.
Remark . The fuzzy chromatic polynomial of fuzzy graph with fuzzy vertex set and fuzzy edge set, ( , ), ≥ will be decreased when the value of will increase, since in fuzzy graph with fuzzy vertices and fuzzy edges, both |V | and |E | decrease as increases (see Table 1).
Example . Consider a fuzzy graph with fuzzy vertex set and fuzzy edge set in Figure 6.
By routine computation, the fuzzy chromatic polynomial of the fuzzy graph in Figure 6 is obtained as 6 Advances in Fuzzy Systems   Table 1 shows that the fuzzy chromatic polynomial of the fuzzy graph in Example 33 deceases as increases (see column 1 and columns 6-9 of Table 1). Moreover, both the number of vertices and the number of edges of -cut graph of the fuzzy graph decrease as increases (see column 1 and columns 2 and 3 of Table 1).

Main Results
In this section, we prove some relevant results on fuzzy chromatic polynomial for fuzzy graphs which are discussed in Section 3. Moreover, the fuzzy chromatic polynomial of complete fuzzy graph and the fuzzy chromatic polynomial of fuzzy graphs which are cycles and fuzzy cycles are studied.  The relations between the -cut graph of a fuzzy graph and the value of = 0, the fuzzy chromatic polynomial of a fuzzy graph, and the chromatic polynomial of corresponding complete crisp graph can be determined below.   The following is an important theorem for computing the fuzzy chromatic polynomial of a fuzzy graph explicitly. Case . Let = 0. Then by Theorem 39, the fuzzy chromatic polynomial of the fuzzy graph is the chromatic polynomial of a complete crisp graph with n vertices, . That is, Case . Let = , where = min( ). Then by Theorem 40, the fuzzy chromatic polynomial of the fuzzy graph is the chromatic polynomial of its underlying crisp graph * . That is, Case . Let < ≤ 1; since is in I, then by Definition 20 ( , ) = ( , ). This completes the proof.
The following result shows that the degree of fuzzy chromatic polynomial of a fuzzy graph is less than or equal to the number of vertices of the fuzzy graph. It is a significant difference from crisp graph theory.
Proof. Let us prove the theorem by considering two cases.
. . Complete Fuzzy Graph and Fuzzy Cycle. In 1989, Bhutani [11] introduced the concept of a complete fuzzy graph as follows.
The following result gives the underlying crisp graph of a complete fuzzy graph is a complete crisp graph.
Lemma 44. Let be a complete fuzzy graph and * be its underlying crisp graph. en * is a complete crisp graph.
It is clear from the following example.
Example . We show that the underlying crisp graph of a complete fuzzy graph is complete crisp graph. Let Case . Let = 0, and since is a fuzzy graph with n vertices then, by Theorem 39, we have ( , ) = ( , ).
Case . Let < ≤ 1; since is in I and is a fuzzy graph then by Theorem 41 ( , ) = ( , ). This completes the proof.
Advances in Fuzzy Systems 9 The fuzzy chromatic polynomial of a complete fuzzy graph with | | = | | can be defined in terms of the chromatic polynomial of complete crisp graph as follows.
If | | = | | = , ≥ 1, then the fuzzy chromatic polynomial of a complete fuzzy graph is defined as where , ≥ 1 is a complete crisp graph with n vertices. Note that the advantage of Definition 49 is that the fuzzy chromatic polynomial of a complete fuzzy graph with | | = | | can be obtained without using the formal procedures. This situation can be illustrated in the following example.
A fuzzy graph is a cycle and a fuzzy cycle as defined in [9,32].
Remark . If a fuzzy graph is cycle with n vertices, then * = , where is a cycle crisp graph with n vertices.
The following theorem is important for computing the fuzzy chromatic polynomial of a fuzzy graph which is cycle. Case . Let = 0, and since is a fuzzy graph with n vertices then by Theorem 39, we have ( , ) = ( , ).
Case . Let = , where = min( ); since is a fuzzy graph then, by Theorem 40, we have and since G is cycle with n vertices, then, by Remark 52 and (15), we get ( , ) = ( , ).
Case . Let < ≤ 1; since is in I and is a fuzzy graph then, by Theorem 41, ( , ) = ( , ). Hence, the result holds. Conversely, we shall prove it by contrapositive. Suppose is not a cycle. Then, by Definition 51, * is not a cycle. This implies that * ̸ = . Hence, Therefore, if (14) is true, then is a cycle. This completes the proof.
Corollary 54. Let be a fuzzy graph. If is a fuzzy cycle with vertices; then ( ) is true. Proof. Let be a fuzzy cycle with vertices. Since is fuzzy cycle, then its * is cycle. Then by Definition 51, is a cycle. Then by Theorem 53, the result is immediate.
Remark . The converse of Corollary 54 is not true. It can be seen from the following example.
Example . Consider the fuzzy graph given in Figure 8.
Remark . In general, the meaning of fuzzy chromatic polynomial of a fuzzy graph depends on the sense of the index . It can be interpreted that for lower values of more numbers of different k-colorings are obtained. On the other hand, for higher values of less number of different k-colorings are obtained. Therefore, the fuzzy chromatic polynomial sums up all this information so as to manage the fuzzy problem.

Conclusions
In this paper, the concept of fuzzy chromatic polynomial of fuzzy graph with crisp and fuzzy vertex sets is introduced. The fuzzy chromatic polynomial of fuzzy graph is defined based on -cuts of the fuzzy graph. Some properties of the fuzzy chromatic polynomial of fuzzy graphs are investigated. The degree of fuzzy chromatic polynomial of a fuzzy graph is less than or equal to the number of vertices in fuzzy graph which is a significant difference from the crisp graph theory. Further, the fuzzy chromatic polynomial of fuzzy graph with fuzzy vertex set will decrease when the value of will increase. Also, more results on fuzzy chromatic polynomial of fuzzy graphs have been proved. The relation between the fuzzy chromatic polynomial of fuzzy graph and its underlying crisp graph and the relation between the fuzzy chromatic polynomial of fuzzy graph and complete crisp graph are established. Finally, the fuzzy chromatic polynomial for complete fuzzy graph and fuzzy cycle is studied. Also, the fuzzy chromatic polynomial of complete fuzzy graph is defined explicitly when | | = | |.
We are working on fuzzy chromatic polynomial based on different types of fuzzy coloring functions as an extension of this study.

Data Availability
The data used to support the findings of this study are included within the article.