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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

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 [

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. [

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 using

The rest of paper is organized as follows. In Section

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

In this subsection some basic definitions and concepts of vertex coloring and chromatic polynomials are reviewed [

Let

If

The smallest

Let

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.

Let

In this subsection, some basic definitions on fuzzy set and fuzzy graphs are reviewed [

A fuzzy set

A fuzzy graph

In this paper, we denote

Note that a fuzzy graph is a generalization of crisp graph in which

The fuzzy graph

The fuzzy graph

For any fuzzy graph

The underlying crisp graph

For

The concept of chromatic number of fuzzy graph was introduced by Munoz et al. [

If

By this definition the chromatic number of fuzzy graphs G is the fuzzy number

Later Eslahchi and Onagh [

A family

for every strong edge

The minimum number

Incorporating the features of the above two definitions, Kishore and Sunitha [

For each

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

Crisp vertex coloring and the chromatic polynomial of

Let

The number of distinct k-coloring on the vertices of

Let

Fuzzy chromatic polynomial of a fuzzy graph is defined as follows.

Let

That is,

In this subsection, we present fuzzy chromatic polynomial of fuzzy graphs with crisp vertices and fuzzy edges.

A fuzzy graph

A fuzzy graph is defined as a pair

V is the crisp set of vertices (that is,

the function

Let

Consider the fuzzy graph

The fuzzy graph

In

Different fuzzy chromatic polynomials of the fuzzy graph

The fuzzy chromatic polynomial depends on the values of

For the fuzzy graph

The fuzzy chromatic polynomial of a fuzzy graph does not need to be decreased when the value of

Let

Consider the fuzzy graph

A fuzzy graph

A crisp graph

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.

Let

Let

In Section

Consider a fuzzy graph

A fuzzy graph

In

Different fuzzy chromatic polynomials of the fuzzy graph G in Example

The fuzzy chromatic polynomial varies for the same fuzzy graph

For fuzzy graph

For instance, for

The fuzzy chromatic polynomial of fuzzy graph

The values of

| | | | | | | | |
---|---|---|---|---|---|---|---|---|

0 | 3 | 3 | 3 | k(k-1)(k-2) | 6 | 24 | 60 | 120 |

0.1 | 3 | 3 | 3 | k(k-1)(k-2) | 6 | 24 | 60 | 120 |

0.5 | 3 | 2 | 2 | k(k-1)^{2} | 2 | 12 | 36 | 80 |

0.7 | 2 | 1 | 2 | k(k-1) | 2 | 6 | 12 | 20 |

0.8 | 2 | 0 | 1 | k^{2} | 1 | 4 | 9 | 16 |

1 | 1 | 0 | 1 | k | 1 | 2 | 3 | 4 |

Consider a fuzzy graph

A fuzzy graph

By routine computation, the fuzzy chromatic polynomial of the fuzzy graph

Table

In this section, we prove some relevant results on fuzzy chromatic polynomial for fuzzy graphs which are discussed in Section

The relations among different

Let

Let

Let

Let

Let

Let

The relations between the

Let

Let

For

Let

Let

The relation between fuzzy chromatic polynomial of a fuzzy graph and the chromatic polynomial of corresponding underlying crisp graph is established below.

Let

Let

The following is an important theorem for computing the fuzzy chromatic polynomial of a fuzzy graph explicitly.

Let

Let

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.

Let

Let us prove the theorem by considering two cases.

In 1989, Bhutani [

A complete fuzzy graph is a fuzzy graph

The following result gives the underlying crisp graph of a complete fuzzy graph is a complete crisp graph.

Let

It is clear from the following example.

We show that the underlying crisp graph of a complete fuzzy graph is complete crisp graph. Let

For complete fuzzy graph

The following is an important theorem for computing the fuzzy chromatic polynomial of a complete fuzzy graph explicitly.

Let

Let

For a complete fuzzy graph

The fuzzy chromatic polynomial of a complete fuzzy graph with

Let

If

Note that the advantage of Definition

Consider the complete fuzzy graph

A complete fuzzy graph

In Figure

Since

Let

If a fuzzy graph

The following theorem is important for computing the fuzzy chromatic polynomial of a fuzzy graph which is cycle.

Let

Let

Conversely, we shall prove it by contrapositive. Suppose

Let

Let

The converse of Corollary

Consider the fuzzy graph

A fuzzy graph

In Figure

In general, the meaning of fuzzy chromatic polynomial of a fuzzy graph depends on the sense of the index

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

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

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

Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.