Neutrosophic Fuzzy Bi magic Labeling of Fan Graphs

Neutrosophic Fuzzy Bi magic Labeling of Fan Graphs

Dr. N. Sarala (1) and G. Vidhya (2)

(1) Associate professor and (2) Research Scholar

(1,2) Department of Mathematics, A.D.M. College for Women (Autonomous), Affiliated to Bharathidasan University, Nagapattinam, Tamil Nadu, India.

Abstract – In order to depict the ambiguity and uncertainty in graph structures, this work presents the idea of Neutrosophic Fuzzy Bi Magic Labeling Graphs, which blends fuzzy set theory and neutrosophic logic.We define NFML graphs and investigate their characteristics, such as neotroscopic fuzzy indices and bi magic labeling criteria. Neutrosophic fuzzy bimagic labeling of a fan graph is investigated in this article.

Key words: Fuzzy Labeling, Neutrosophic fuzzy Labeling, Bi magic Labeling, Fan graph.

1. INTRODUCTION

   Graph Theory is a fundamental tool for modeling complex systems and relations in various fields, including computer science, engineering, and social networks. However, real world systems often involve uncertainty, ambiguity, and imprecision, which can be challenging to represent using traditional graph theory. To address this limitation, fuzzy graph theory and magic labeling have been introduced. Fuzzy labeling is a technique used to represent uncertainty and imprecision in graph structures. In fuzzy labeling, edges and vertices are assigned fuzzy lables, which are typically represented as fuzzy sets. This allows for a more nuanced representation of relations between nodes, enabling the modeling og complex systems with uncertain or imprecise connections. Magic labeling is a specific type of graph labeling where the sum of labels of edges and vertices satisfies certain conditions. The combination of fuzzy labeling and magic labeling enables the representation of complex systems with uncertain relationships and specific structural properties. Neutrosophic fuzzy labeling is an extension of traditional fuzzy labeling, incorporating neutrosophic logic to represent uncertainty, ambiguity and imprecision in graph structures. Neutrosophic logic, introduced by Florentin Samarandache. In Neutrosophic fuzzy labeling, each label consist of three components Truth, Indeterminacy, Falsity, allowing for a more nuanced representation of complex relationships and uncertain information. This approach enables the modeling of realworld systems with inherent ambiguity, uncertainty and imprecision.

2. PRELIMINARIES

   Definition.2. 1

   Let C* = (V, E) be a simple graph. Then C* = (, ) is called a fuzzy graph on C*, if : V [0, 1] and : E

   [0, 1] and for all x, y E , (x, y) min [(x), (y)] .A fuzzy graph C= (, )) on C* is called a fuzzy labeling graph if and one to one maps for all x, y .

   Definition. 2.2

   A fuzzy labeling graph C = (, ) on C is called fuzzy magic labeling graph if there exists m, which is called magic value such that for all x, y E.

   (x) + (y) + (x, y) = m

   Definition. 2.3

   If the total membership values of the vertices and edges incident at the vertices are k1 and k2, where k1 and k2 are constants, then a fuzzy labeling graph admits Bi-magic labelling

   Definition. 2.4

   A Neutrosophic Fuzzy Graph C = (, ) where = {v1, v2, v3,. Vn} such that T1: V [0, 1], I1: V [0, 1] and F1: V [0, 1] denote the degree of truth membership, indeterminacy membership and falsity membership of the element vi V ( i=1,2n) , E V× V where

   T2: V× V [0, 1], I2: V× V [0, 1], and F2: V× V [0, 1] are such that T2 (vi, vj) min [T1(vi), T1(vj))],

   I2 (vi, vj) min [I1(vi), I1(vj))],

   F2(vi, vj) max [F1(vi), F1(vj))]

   0 T2 (vi, vj) + I2 (vi, vj) + F2 (vi , vj) 3 for every (vi, vj) E (j=1,2. n)

   A neutrosophic fuzzy labeling graph is a Neutrosophic fuzzy magic labeling graph if their exist an M such that M equals

   {T1(vi)+ T1(vj) + T2 (vi, vj),

   [I1(vi) +I1(vj) + I2 (vi, vj),

   F1(vi) + F1(vj) + F2 (vi, vj)}

   Definition. 2.5

   A fan graph is defined as the graph join of an empty graph on m nodes and a path graph on n nodes. This corresponds to usual fan graphs when m=1 and double fan graphs when m=2.

   Definition. 2.6

   A Fan graph with fuzzy labeling is called fuzzy Fan graph. A Fan is a fuzzy graph consists of two node sets F and || =1 and || > 1, such that µ( F,Fi ) >0, where i=1 to n and µ( Fi,Fi+1) >0 where i=1 to n-1.

3. NEUTROSOPHIC FUZZY BI-MAGIC LABELLING OF FAN GRAPH

Preposition:3.1

Every fuzzy fan graph F1, n is a Neutrosophic fuzzy bimagic labeling graph for n=3.

Proof:

Let 1, 2, be the vertices and 1, 2 be the edges of the fan graph.

The vertices are defined for the truth, indeterminacy and false membership functions : Ø [0, 1], : Ø [0, 1], µ: Ø [0, 1] respectively for the vertices are as follows,

( ) = 43, for all n = 1 to 4

41

( ) = 31

4

µ ( ) = 51

10

F1,3 Fan Graph

The vertices of fan graph F1,3 is noted as follows

1(0.33,0.5,0.4),2(0.71,0.63,0.45), 3(0.82,0.67,0.47), 4(0.87,0.69,0.48).

Then the corresponding edge values are 1(0.26,0.42,0.44), 2(0.32,0.23,0.46), 3(0.16,0.17,0.42),

4(0.10,0.36,0.41), 5(0.27,0.21,0.47)

Existence of bi magic values are

1(0.33,0.5,0.4) + 1(0.26,0.42,0.44) + 2(0.71,0.63,0.45) is (1.3,1.55,1.29)

,2(0.71,0.63,0.45) + 2(0.32,0.23,0.46) + 3(0.82,0.67,0.47) is (1.85,1.53,1.38)

3(0.82,0.67,0.47) +3(0.16,0.17,0.42) + 4(0.87,0.69,0.48) is (1.85,1.53,1.37),

4(0.87,0.69,0.48) + 4(0.10,0.36,0.41) + 1(0.33,0.5,0.4) is (1.3,1.55,1.29)

4(0.87,0.69,0.48) + 5(0.27,0.21,0.47) + 2(0.71,0.63,0.45) is (1.85,1.53,1.38)

Therefore the Bi magic values are (1.3,1.55,1.29) and (1.85,1.53,1.38) Hence the Fan graph F1,3 admits a Neutrosophic fuzzy bi magic labeling.

Preposition: 3.2

Every fuzzy fan graph F1, n is a Neutrosophic fuzzy bi magic labeling graph for n=5.

Proof:

Let 1, 2, be the vertices and 1, 2 be the edges of the fan graph.

The vertices are defined for the truth, indeterminacy and false membership functions : Ø [0, 1],

: Ø [0, 1], µ: Ø [0, 1] respectively for the vertices are as follows,

( ) = 3+1, for n = 5,6,8 for some p=31,1i6

3 , for n=9

311 , for n=6,10

( ) =

,for n=1 to n

102

µ ( ) = 7±1 ,for n=7,9

102

7

102

, for n=8

7±3 ,for n= 2,8

102

F1,5 Fan Graph

The vertices F1,5 as follows

The vertices are defined for the truth, indeterminacy and false membership functions : Ø [0, 1],

: Ø [0, 1], µ: Ø [0, 1] respectively for the vertices are as follows,

1(0.61,0.19,0.17),2(0.93,0.14,0.5), 3(0.55,0.15,0.62), 4(0.81,0.16,0.53), 5(0.52,0.17,0.59),

6(0.87,0.18,0.56).

Then the corresponding edge values are 1(0.03,0.13,0.32) , 2(0.07,0.08,0.59), 3(0.19,0.06,0.56),

4(0.22,0.04,0.59), 5(0.16,0.02,0.56), 6(0.09,0.09,0.26), 7(0.44,0.10,0.23), 8(0.15,0.11,0.29),

9(0.41,0.12,0.20)

Existence of bi magic values are

1(0.61,0.19,0.17) + 1(0.03,0.13,0.32) + 2(0.93,0.14,0.5) = (1.57,0.46,0.99)

1(0.61,0.19,0.17) + 9(0.41,0.12,0.20) + 3(0.55,0.15,0.62) = (1.57,0.46,0.99)

1(0.61,0.19,0.17) + 8(0.15,0.11,0.29) + 4(0.81,0.16,0.53) = (1.57,0.46,0.99)

1(0.61,0.19,0.17) + 7(0.44,0.10,0.23) + 5(0.52,0.17,0.59) = (1.57,0.46,0.99)

1(0.61,0.19,0.17) + 6(0.090.09,0.26) + 6(0.87,0.18,0.56) = (1.57,0.46,0.99)

2(0.93,0.14,0.5) + 2(0.07,0.08,0.59) + 3(0.55,0.15,0.62) = (1.55,0.37,1.71)

3(0.55,0.15,0.62) + 3(0.19,0.06,0.56) + 4(0.81,0.16,0.53) = (1.55,0.37,1.71)

4(0.81,0.16,0.53) + 4(0.22,0.04,0.59) + 5(0.52,0.17,0.59) = (1.55,0.37,1.71)

5(0.52,0.17,0.59) + 5(0.16,0.02,0.56) + 6(0.87,0.18,0.56) = (1.55,0.37,1.71). Hence the Fan graph F1,3

admits a Neutrosophic fuzzy bi magic labeling.

CONCLUSION:

The definitions of Neutrosophic fuzzy bi magic labelling for the fan graphs F1,3, F1,5 have been investigated in this paper.

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