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New Concepts of Picture Fuzzy Graphs with Application

Profile image of Dr. Anita PalDr. Anita Pal

2019, Mathematics

https://doi.org/10.3390/MATH7050470
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Abstract

The picture fuzzy set is an efficient mathematical model to deal with uncertain real life problems, in which a intuitionistic fuzzy set may fail to reveal satisfactory results. Picture fuzzy set is an extension of the classical fuzzy set and intuitionistic fuzzy set. It can work very efficiently in uncertain scenarios which involve more answers to these type: yes, no, abstain and refusal. In this paper, we introduce the idea of the picture fuzzy graph based on the picture fuzzy relation. Some types of picture fuzzy graph such as a regular picture fuzzy graph, strong picture fuzzy graph, complete picture fuzzy graph, and complement picture fuzzy graph are introduced and some properties are also described. The idea of an isomorphic picture fuzzy graph is also introduced in this paper. We also define six operations such as Cartesian product, composition, join, direct product, lexicographic and strong product on picture fuzzy graph. Finally, we describe the utility of the picture fuzzy ...

Figures (6)
Figure 2. A strong picture fuzzy graph. An example of a strong picture fuzzy graph is shown in Figure 2.
Figure 2. A strong picture fuzzy graph. An example of a strong picture fuzzy graph is shown in Figure 2.
An example of picture fuzzy graph is shown in Figure 1. Definition 19. A picture fuzzy graph G = (A,B) is said to be regular picture fuzzy graph if Figure 1 is also an example of regular picture fuzzy graph.
An example of picture fuzzy graph is shown in Figure 1. Definition 19. A picture fuzzy graph G = (A,B) is said to be regular picture fuzzy graph if Figure 1 is also an example of regular picture fuzzy graph.
Figure 3. A complete picture fuzzy graph. An example of a complete picture fuzzy graph is shown in Figure 3. Note that every complete picture fuzzy graph is a strong picture fuzzy graph but not conversely.
Figure 3. A complete picture fuzzy graph. An example of a complete picture fuzzy graph is shown in Figure 3. Note that every complete picture fuzzy graph is a strong picture fuzzy graph but not conversely.
Figure 4. A picture fuzzy graph and its complement picture fuzzy graph. An example of complement picture fuzzy graph is shown in Figure 4.
Figure 4. A picture fuzzy graph and its complement picture fuzzy graph. An example of complement picture fuzzy graph is shown in Figure 4.
Figure 5. Undirected picture fuzzy social network.
Figure 5. Undirected picture fuzzy social network.
Table 1. The picture fuzzy degree centrality of research team. Figure 6. Directed picture fuzzy social network. @ 1S an addition operation oF picture ruzzy set. Let Gy = (V, E,7) be a directed picture fuzzy social network of the a research Team, where V = {0}, 02,...,07} represents a collection of seven researchers, Egg represents directed picture fuzzy relation between the seven researchers. This social network is shown in Figure 6. We determine the picture fuzzy in degree, picture fuzzy in degree and picture fuzzy degree centrality of the research term using (23)-(25). This three degree values are shown in Table 1. We use the raking method [44,45] of picture fuzzy set to compare the different degree values. Based on the ranking of picture fuzzy set, research (node) 4 has got highest score value of picture fuzzy in degree centrality. It means that researcher 4 has higher acceptance and good interpersonal relationship in network. The research 2 has got the highest score value of picture fuzzy out degree centrality. It describes that node 2 can nominate many other researchers.
Table 1. The picture fuzzy degree centrality of research team. Figure 6. Directed picture fuzzy social network. @ 1S an addition operation oF picture ruzzy set. Let Gy = (V, E,7) be a directed picture fuzzy social network of the a research Team, where V = {0}, 02,...,07} represents a collection of seven researchers, Egg represents directed picture fuzzy relation between the seven researchers. This social network is shown in Figure 6. We determine the picture fuzzy in degree, picture fuzzy in degree and picture fuzzy degree centrality of the research term using (23)-(25). This three degree values are shown in Table 1. We use the raking method [44,45] of picture fuzzy set to compare the different degree values. Based on the ranking of picture fuzzy set, research (node) 4 has got highest score value of picture fuzzy in degree centrality. It means that researcher 4 has higher acceptance and good interpersonal relationship in network. The research 2 has got the highest score value of picture fuzzy out degree centrality. It describes that node 2 can nominate many other researchers.

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