Operations on Intuitionistic Fuzzy Directed Graphs

Yogeesh N1*, Dr. P.K. Chenniappan2

1 Research Scholar and Assistant Professor of Mathematics, Government First Grade College, Badavanahalli, Tumkur, Karnataka, Calorx Teacher's University, Ahmedabad, Gujarat, India

Email: yogeesh.r@gmail.com

2 Research Guide, Professor & Head, Department of Mathematics, Calorx Teacher's University, Ahmedabad, Gujarat, India

Abstract- The motivation for introducing IFGs is due to The first definition and concept of Intuitionistic Fuzzy Graph (IFG) was introduced. Karunambigai analyzed the properties of minmax IFGs in Shannon and Atanassov defined intuitionistic fuzzy graphs using five types of Cartesian products. In this paper IFGs so defined are named. Some isomorphic properties on IFGs are discussed. et al., discussed the properties of isomorphism on fuzzy graphs and properties of isomorphism on strong fuzzy graphs in which motivated us to develop the same on IFGs and on strong IFGs. The main aim of this study is to build basic definitions of an IFG which will be useful for the researchers for their future study in IFGs. Since the title of the paper is given so. For graph theoretical definitions and throughout this paper all the properties are analyzed for simple minmax IFG.

Keywords: Intuitionistic, Fuzzy Graphs, Fuzzy Directed Graph

INTRODUCTION

Graph theory is a very beneficial device in fixing combinatorial problems in exceptional areas consisting of geometry, algebra, wide variety principle, topology, operations studies, optimization, laptop technological know-how, engineering, and bodily, organic, and social structures. Point-to-point interconnection networks for parallel and allotted systems are normally modeled by means of directed graphs (or digraphs). A digraph is a graph whose edges have instructions and are referred to as arcs (edges). Arrows on the arcs are used to encode the directional information: an arc from vertex (node) 𝑥 to vertex 𝑦 suggests that one may circulate from 𝑥 to 𝑦 but no longer from 𝑦 to 𝑥. Presently, technology and era are featured with complicated processes and phenomena for which entire information isn't always continually to be had. For such instances, mathematical fashions are advanced to handle styles of structures containing elements of uncertainty. A large quantity of those models are based totally on an extension of the ordinary set principle, namely, fuzzy units. The notion of fuzzy units turned into brought through Zadeh [1] as a way of representing uncertainty and vagueness. Since then, the theory of fuzzy units has end up a vigorous place of studies in specific disciplines, such as clinical and life sciences, management sciences, social sciences, engineering, data, graph theory, synthetic intelligence, sign processing, multiagent systems, pattern reputation, robotics, laptop networks, professional systems, selection making, and automata concept.

Fuzzy graph idea is locating an increasing number of packages in modeling actual time systems where the extent of statistics inherent in the gadget varies with unique stages of precision. Fuzzy models are getting useful because of their aim of reducing the variations between the conventional numerical fashions used in engineering and sciences and the symbolic fashions used in expert systems.

Kauffman’s initial definition of a fuzzy graph was based totally on Zadeh’s fuzzy family members. Rosenfeld brought the fuzzy analogue of numerous fundamental graph-theoretic principles and Bhattacharya gave a few comments on fuzzy graphs. Mordeson and Nair [6] defined the idea of complement of fuzzy graph and studied some operations on fuzzy graphs. In , the definition of supplement of a fuzzy graph turned into modified in order that the supplement of the supplement is the unique fuzzy graph, which is of the same opinion with the crisp graph case. Atanassov introduced the concept of intuitionistic fuzzy members of the family and intuitionistic fuzzy graphs. Akram et al. added many new concepts, inclusive of sturdy intuitionistic fuzzy graphs, intuitionistic fuzzy hyper-graphs, intuitionistic fuzzy cycles, and intuitionistic fuzzy trees. Wu mentioned fuzzy digraphs. In this paper, the intuitionistic fuzzy organizational neural network models, intuitionistic fuzzy neurons in medical diagnosis, intuitionistic fuzzy digraphs in vulnerability evaluation of gasoline pipeline networks, and intuitionistic fuzzy digraphs in tour time are supplied as examples of intuitionistic fuzzy digraphs in selection assist systems. Algorithms of those selection support structures also are designed and applied.

OBJECTIVES

  1. To investigate edge dominance in fuzzy graphs and intuitionistic fuzzy graphs in a safe and fair manner.

METHOD ANALYSIS

Consider the two IFDGs   and , where  and  are the vertex sets and,  and  are the edge sets of  and  respectively.

Definition 1.

The addition of two IFDGs  and , denoted by , is defined by

 , where

and

Example 2.

Consider the IFDGs  and  as in Figure 2.

Figure 2.:  and

The index matrix of  is , where

and

The index matrix of  is

, where   and

The index matrix of  is ,

Where,   and

The graph of  is shown in Figure 3.

Figure 3:

Definition 3

The vertex wise multiplication of two IFDGs  and , denoted by , is defined by ,

Where,

 if  ; and .

Example 4

The index matrix of  is ,

Where,  and

The graph of , a null IFDG, is displayed in Figure 5.

Figure 5 :

Example 5

Consider the two IFDGs  and  as shown in Figure 6.

Figure 6:  and

Figure 6 depicts , which is not a null IFDG.

Figure 7:

Definition 6

 is the structural subtraction of two IFDGs  and , indicated by .

, where  is the set theoretic difference operation and

 for  and , if , then graph of  is also empty. Figure 7 shows the structural subtraction of the IFDGs  and  given in Figure 2.

Figure 8:

CONCLUSION

On this paper, the definition of complement of  an IFG is given and a few properties of self - complementary IFGs are studied. Also, we bear in mind the operations union,  be part of, Cartesian  product and  composition of IFGs and proved that  the supplement  of the join  of two  IFGs is the  union of  their enhances. Also, we've proved that composition of two strong IFGs is also sturdy.  Much greater  work  can be  executed to  investigate the  shape of  IFGs which could  have  programs  in  Communication  networks,  Information Technology,  Pattern  Clustering,  Image  Retrieval  and  so  on.  The  authors proposed to similarly extend the principles of fuzzy graphs into IFGs.

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