fuzzy soft expert graph.

The concept of fuzzy set theory was introduced by Zadeh [1] to solve difficulties in dealing with uncertainties. Since then, the theory of fuzzy sets and fuzzy logic have been examined by many researchers to solve many real-life problems involving ambiguous and uncertain environments. Atanassov [2] introduced the concept of intuitionistic fuzzy sets as an extension of Zadeh's [1] fuzzy set. Molodtsov [3] introduced the concept of soft set theory as a new mathematical tool for dealing with uncertainties. Molodtsov's [3] soft sets give us a new technique for dealing with uncertainty from the viewpoint of parameters. Maji et al. [4] proposed soft sets. Alkhazaleh and Salleh [5], [6] defined the concept of a soft expert set. Graph theory has now become a major branch of applied mathematics and is generally regarded as a branch of combinatorics. The graph is a widely used tool for solving combinatorial problems in different areas, such as geometry, algebra, number theory, topology, optimization, and computer science. When the relations between nodes (or vertices) in problems are indeterminate, the fuzzy graphs and their extensions [7–13]. The above theories have been applied to many areas, including real decision-making problems [14-32].

Using examples to make the concept easier, we have discussed different operations defined on intuitionistic fuzzy soft expert graphs. Isotonic fuzzy soft expert graphs are pictorial representations in which each vertex and edge is an element of intuitionistic fuzzy soft sets. This paper has been arranged as the following.

Section 2 presents some basic concepts about graphs and fuzzy soft sets, which will be employed in later sections. Section 3 presents the concept of intuitionistic fuzzy soft expert graphs, and some of their fundamental properties have been studied. In Section 4, we present an application of intuitionistic fuzzy soft expert graphs in decision-making, and then an illustrative example is given. Finally, the conclusions are drawn in Section 5.

Definition 1 ([33]). A fuzzy graph is a pair of functions H where is a fuzzy subset of a non-empty set V, and is a symmetric fuzzy relation on . i.e. 6 and 6 6 such that no n o for all n o 6 where no denotes the edge between and o and n o denotes the minimum of n and o . is called the fuzzy vertex set of V, and is called the fuzzy edge set of E.

Definition 2 ([33]). The fuzzy subgraph I is called a fuzzy subgraph of H , if n n for all n 6 and n o n o for all n o 6.

Definition 3 ([34]). An intuitionistic fuzzy graph is of the form H 6 where I. 6 o o o such that 6 and 6 denote the degree of membership and non membership of the element o 6 respectively, and o o for every o 6 b g

and for every .

Let be a universe, a set of parameters, a set of experts (agents), and bl a set of opinions. Let and .

Definition 4. Let H be a simple graph, and be the set of parameters. Let G3 be the set of all intuitionistic sets in . By an intuitionistic fuzzy soft expert graph IFSEG, we mean a 4-tuple H

r r r r are intuitionistic fuzzy soft sets over and respectively, such that r fbg r r fbg r

For all r and . We can also denote an IFSEG by H H G3 which is a parameterized family of graphs G3 we call them intuitionistic fuzzy soft expert graphs.

Example 1. Suppose that H be a simple graph with be a set of parameters and i be a set of experts. An IFSEG is given in Table 1 below and and for all and for all l

Example 2. Suppose that H be a simple graph with be a set of parameters and i be a set of experts. An IFSE subgraph of Example 1 is given in Table 2 below and and for all and for all

Definition 6. An G3 subgraph H H is said to be spanning G3 subgraph of H H if for all Definition 7. An agree-G3 graph H H over H is an G3 subgraph of H H defined as follows:

Example 3. Consider Example 1, then, the agree-G3 graph H H over H .

Example 4. Consider Example 1, then the disagree-G3H H H over H .

H with where the membership and non membership of the union are as follows:

b g h b g h

b g h b g h

f b g hmapbl b g h p b g h

b g h p b g h

fbg p p b g hmapbl b g r h r b g r h b g r b g r h r r b g r h b g r b g r f r r

hmapbl b g r h p r b g r h b g r b g r h r p r b g r h b g r b g r fbg p r p r

hmapbl Proposition 1. The union H H of two G3Hl H H and H H is an G3H.

fbg r , so r fbg r .

So r fbg r .

Now p r p r f p p r f p p r .

So p r f p p r .

, we can show the same as done above.

r f r

f fbg r fbg r

fbg f f r r fbg r

r fbg p r

fbg f p r f r

f fbg fbg r r Hence, the union H H H is an G3H.

Definition 10. The intersection of two G3Hl H H and H H is denoted by

are as follows:

fbg b

fbg r r b

f r r b

Proposition 2. The intersection H H of two G3Hl H H and H H is an G3H where

I. b , then r r fbg r fbg r ,

so r fbg r .

Now r r f r f r , so r f r . Similarly, b we can show the same as done above.

r f r

fbg fbg r fbg r

fbg fbg fbg r r fbg r

Now

r f r

f f r f r

f f f r r

Hence the intersection H H H is an G3H.

Assume that a company wants to fill a position to be chosen by an expert committee. Suppose that H be a simple graph with 9 be a set of parameters for computer knowledge. Let 8 i be a set of two expert committee members. An G3H is given in Table 5 below and and for all and for all After serious deliberation, the committee constructs the following G3H.

I. Input the G3H.

II. Find the mean of each G3 edge according to the relationship among criteria for each alternative.

III. Find an agree-G3H and a disagree- G3H.

IV. Find D for agree- G3H.

V. Find for disagree- G3H.

VI. Find 3 D .

VII. Find for which l fl where,l is the optimal choice object. If r has more than one value, then the company could choose any one of them using its option.

G3 edges according to the relationship among criteria for each alternative.

Table 7 presents the agree-G3H by using the mean of each G3H.

Table 8 presents the disagree-G3H respectively, by using the mean of each G3H.

Since max 3 hence the committee will choose candidate with a master's degree for the job.

In this paper, we have introduced the concept of IFSEG, union and the intersection of them has been explained with an example which has broader application in the field of modern sciences and technology, especially in research areas of computer science, including database theory, data mining, neural networks, expert systems, cluster analysis, control theory, and image capturing.

M. S. research design, conceptualization, and validation. V. U. data gathering, computing, and editing. The authors have read and agreed to the published version of the manuscript.

The authors declare that no external funding or support was received for the research.

All data supporting the reported findings in this research paper are provided within the manuscript.

The authors of this article are members of the UDA editorial board and did not participate in the editorial review or decision to publish this work. They have declared that they have no conflicts of interest related to this work.