#### Abstract

The article describes a new thought on the chromatic polynomial of an intuitionistic fuzzy graph which is illustrated based on -level graphs. Besides, the alpha-beta fundamental set of an intuitionistic fuzzy graph is also defined with a vivid description. In addition to that, some characterizations of the chromatic polynomial of an intuitionistic fuzzy graph are specified as well verified. Furthermore, some untouched properties of the -level graph are also projected and proved.

#### 1. Introduction

Graph coloring has a lot of applications in areas such as wireless radio channel assignment [1], timetable scheduling [2], and job scheduling [3]. Graph coloring is described as vertex coloring, edge coloring, and map coloring. The concept of the chromatic polynomial was introduced as the number of distinct ways of performing map coloring [4]. The chromatic polynomials of many families of graphs have been computed so far [5, 6]. In addition, the computations of the chromatic polynomials of certain graphs using the Mobius inversion theorem were reported by Srinivasa Rao et al. [7].

The fuzzy set theory was introduced by Zadeh [8]. A fuzzy set communicates an element of a given set with a membership value in [0, 1]. The introduction of fuzzy graph theory was given based on the fuzzy set introduced by Kauffman [9]. The traditional fuzzy set cannot be used to completely describe all evidence in the problems where someone wants to know how much nonmembership values. Such a problem got a solution by Atanassov who introduced an intuitionistic fuzzy set (IFS) [10]. IFS is an extension of Zadeh’s set theory and is described by a membership function, a nonmembership function, and a hesitation function [10]. The concept of an intuitionistic fuzzy graph (IFG) was introduced by Atanassov [11]. Atanassov and Shannon developed certain properties of IFGs [12]. And also, Akram and Davvaz discussed more properties of IFGs [13]. Akram et al. presented strong IFGs and intuitionistic fuzzy line graphs [14]. Akram et al. presented an algorithm for computing the sum distance matrix, eccentricity, radius, and diameter of IFG [15]. Electoral systems [16], human cells clustering [17], and water supply systems [18] are some application areas of IFGs. Also, IFG digraph is applied in medical diagnosis, gas pipelines, and decision-making systems [19].

The chromatic polynomials of fuzzy graphs using *α*-cuts and their algebraic properties were developed by Abebe and Srinivasa Rao Repalle [20]. The coloring of IFG using -cuts (levels) was introduced by Mohideen and Rifayathali [21]. Some properties of IFG by -levels were presented by Akram [22]. *To fill the gap in these articles, the* authors aimed to find the chromatic polynomial of IFG using *-levels, define*-fundamental set, and *discuss more untouched properties of*-*level graphs.* This paper presents the chromatic polynomial of an IFG using -levels by illustrating it based on -level graphs. In addition, it defines the alpha-beta fundamental set of an IFG. Furthermore, it states and proves some properties of the -level graph and the chromatic polynomial of IFG.

This manuscript is organized as follows: Section 2 is the preliminaries that are essential for understanding the article. Section 3*develops some properties of**-levels of IFGs and proves wherever verification is required.* Section 4 defines the notion of the chromatic polynomial of an IFG using -levels and also develops some related concepts on a chromatic polynomial of an IFG. Section 5 is the summary of the article.

#### 2. Preliminaries

This section defines related concepts such as chromatic polynomial, IFG, and level graph of IFG. In addition, it provides some useful propositions.

*Definition 1 (see [20]). *The number of distinct ways of attaining a proper vertex coloring of a simple graph with at most colors is called chromatic polynomial of and is denoted by .

Proposition 1 (see [23]). *The chromatic polynomial of a simple graph may be found by the formula: , where is the chromatic polynomial of a subgraph of whose one of its edges has been removed, and is the chromatic polynomial of a subgraph of whose one of its edge has been contracted.*

Proposition 2 (see [23]). *Let be a graph containing nonadjacent vertices and , and let be the graph obtained from the contracting edge . Then, .*

Proposition 3 (see [23]). *If are pairwise disjoint components of a simple graph whose union is , then its chromatic polynomial is computed as*

*Definition 2 (see [24]). *An intuitionistic fuzzy graph is of the form , where(i) such that and denote the degree of membership and the degree of nonmembership of the element in , respectively, and , for every , (i) where and are such that Satisfy the constraint, for every Where and are the degree of membership and the degree of nonmembership of the element in respectively.

*Definition 3 (see [21]). *The chromatic number of IFG: is the intuitionistic fuzzy number , where , , , and .

*Definition 4. *[22]. Let be an IFG. Then, for any , -level graph of is defined as the crisp graph , where , and , .

#### 3. The -Fundamental Set and the Properties of -Level Graph of IFGs

This section defines the -fundamental set and illustrates the -level graph of IFGs. In addition, it states and proves some character of the -level graph of IFGs.

*Definition 5. *Let be an IFG. If is such that and for each , then is said to be a fundamental set of .

*Example 1. *Consider an IFG with four intuitionistic fuzzy vertices and five intuitionistic fuzzy edges given in Figure 1.

The fundamental set of an IFG in Figure 1 is {(0, 1), (0.2, 0.6), (0.3, 0.6), (0.4, 0.5), (0.5, 0.5), (0.8, 0.2)}. To show that the reason why (0.2, 0.5) , consider the levels (0.2, 0.6), (0.2, 0.5), and (0.3, 0.6). Now, let us determine if a level (0.2, 0.5) satisfies the condition in the definition of fundamental set. It is forward that 0.2 0.2, 0.6 0.5, 0.2 0.3 but 0.5 0.6. This shows that (0.2, 0.5) .

The level graphs along with their chromatic number are illustrated in Figure 2.

*Remark 1. *(i)Table 1 contains the vertex set, edge set, and chromatic number of for each in (ii)If there is a vertex with , then -level graph exists(iii) is as contains all the vertices listed in and implies that all the vertices are joined(iv)The chromatic number of an IFG in Figure 1 is given by

##### 3.1. *The Properties of*-Level Graph of IFGs

Theorem 1. *Let be an IFG and and be level graphs of with and . Then,*

*Proof. *Let be an IFG. Suppose and are level graphs of with and . Now, and have vertex sets; and , respectively. To figure out the relation has with , we draw them on the same plane as follows.

Based on Figure 3, is an area of a polygon EFGB and is an area of a polygon ABCD. This shows that is contained in . Thus, .

Similarly, and have the edge sets; and , respectively. For comparison, these sets can be interpreted on the same plane as follows.

Based on Figure 4, is an area of a polygon HLNO and is an area of a polygon HIJK. This indicates that is contained in . Thus, .

Alternate proof: Let and be level graphs of with and . Then,(i)The vertex sets to these levels are and . Now, let . Then, , and since and , it implies . This indicates . Thus, (ii)The edge sets of and are and , respectively.Now, let . Then, and and and , and it implies This shows . Hence, .

Corollary 1. *Let and be level graphs of an IFG, with and . Then,*(i)*(ii)*

*Proof. *Assume an IFG, . Let and be two-level graphs of with and . Then, by applying Theorem 1: and . Thus, and hold.

Theorem 2. *Let and be level graphs of an IFG; with and . Then, is a super graph of .*

*Proof. *Let be an IFG and and be given. Then, by Theorem 1, and . From the fact that if the vertex set and edge set of a certain graph are a subset of a vertex set and edge set of another graph , respectively, then is a super graph of . Thus, is a super graph of .

#### 4. The Chromatic Polynomial of IFG Using -Levels

Here, we define a new idea of the chromatic polynomial of IFG using -levels, and also, we compute the chromatic polynomial of an IFG using -levels for illustration. Furthermore, we state and prove some properties of the chromatic polynomial of IFG using -levels.

*Definition 6. *Let be IFG and be its fundamental set. Then, the chromatic polynomial of using -level with at most given colors is denoted by and is defined as the chromatic polynomial of , where . That is, for each .

*Example 2. *Consider the IFG, in Figure 1. All the level graphs of have been put in Figure 2, and by Definition 6, the chromatic polynomial of is the chromatic polynomial of these level graphs. Thus, the chromatic polynomial of is the chromatic polynomial of the level graphs combined in a single equation and is computed as follows:(i)When and , we have , and as , is the chromatic polynomial of . Hence, .(ii)When and , we compute by the addition-contraction method as in Figure 5. Thus, (iii)When and , we have , and it is the chromatic polynomial of . Therefore, .(iv)When and , we obtain and is the chromatic polynomial of . Therefore, .(v)When and , we obtain and is the chromatic polynomial of . Therefore, .(vi)When and , we obtain and is the chromatic polynomial of . Therefore, .In general, the chromatic polynomial of IFG in Figure 1 is summarized in the following equation:

*Remark 2. *(i)As increases the degree of the chromatic polynomial, the chromatic number, the number of edges, and the number of vertices decrease.(ii)As decreases the degree of the chromatic polynomial, the chromatic number, the number of edges, and the number of vertices also decrease.

##### 4.1. Properties of the Chromatic Polynomial of IFGs Using -Levels

Theorem 3. *Let be IFG on intuitionistic fuzzy vertices and let be -level graph of . If and , then*

*Proof. *Let be an IFG. Then, when and , possesses and .

For every edge in , and . So, every edge in satisfies . In addition to edges in , contains edge(s) correspond to edge(s) labeled of . Therefore, every pair of vertices is joined. , as every vertex in , could satisfy . This implies that and represent the same graph. Hence,

Theorem 4. *Let be IFG. If and are intuitionistic fuzzy levels of with and , then*

*Proof. *Assume that IFG. Let and be the level graphs of with and . Then, and . It is obvious to realize that the degree of the chromatic polynomial of a simple graph equals the number of its vertex set. That is, deg( and deg(, and also, since and , (see Corollary 1). Hence, .

Theorem 5. *Let be an IFG and be its underlying graph. If and for each and , then for some 1, 2, ..., m.*

*Proof. *Suppose that is an underlying graph of IFG . Let be a fundamental set of and let and for each and for some . By Theorem 11, is a super graph of for each . This shows that contains all vertices and all edges of as is the underlying crisp graph of . Hence, and represent the same graph, and both have the same chromatic polynomial. That is, .

Theorem 6. *Let and be components of an IFG, , and let be any level such that and . Then,*

*Proof. *Let be IFG and be any of its levels such that and . Also, let and be an intuitionistic fuzzy component of . Now, since and are components of , and are also components of . So, .

Thus, .

Corollary 2. *Let be intuitionistic fuzzy components of an IFG, , and -be any level of . Then,**Proof of this corollary is omitted as it can be forwarded by extending Theorem 6.*

Theorem 7. *If and are intuitionistic fuzzy component graphs of an IFG, with intuitionistic fuzzy vertices and intuitionistic fuzzy vertices, respectively, and is a complete graph on vertices, then .*

*Proof. *Consider an IFG, . Now, let and be intuitionistic fuzzy component graphs of containing *n* and intuitionistic fuzzy vertices, respectively. Then, -level graph of , possesses(i) . , as every vertex in both and , satisfy (ii)Every edge joining any two vertices and in both component graphs and satisfy and . And also, in addition to the edges in both and , contains an edge(s) corresponding to edge(s) labeled of which are actually neither in the crisp graph of nor in the crisp graph of .

From (i), -level graph of has vertices, and from (ii), every pair of vertices of is joined. In other words, the -level graph of is . Hence, .

Corollary 3. *Let be IFG. If and are intuitionistic fuzzy component graphs of with intuitionistic fuzzy vertices and intuitionistic fuzzy vertices, respectively, then*

*Proof. *Based on Theorem 7, -level graph forms a complete crisp graph; and . Thus, the corollary holds.

#### 5. Conclusions

To conclude, the coloring of the IFG using -levels is applied in traffic light control, register allocation, clustering analysis, and decision-making system. The chromatic polynomial of IFG using -levels provides alternative solutions in such application areas. In this work, the notion of the -fundamental set of an IFG has been clearly stated with relevant examples. Apart from that, some characteristics -level graphs have been detailed and proved. Also, the chromatic polynomial of IFG, by using -levels, has been presented. Moreover, the chromatic polynomials of IFG of the different -levels have been derived and compared. Furthermore, certain properties of the chromatic polynomial of an IFG have also been discussed and illustrated in a detailed note.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest concerning the publication of this paper.

#### Authors’ Contributions

All the authors contributed an equal share in the preparation of this article. The authors read and approved the final manuscript.