Fuzzy Graph S

Vol 11, Issue 6,June/ 2020

ISSN NO: 0377-9254 LABELING ON M- FUZZY GRAPH S. Shyamala Malini1 and A. Prathika2 1 Assistant Professor,, Department of Mathematics, St. John’s College, Palayamkottai, Tamil Nadu, India, [email protected] 2 Assistant Professor,, Department of Mathematics, Thiruvalluvar College, Papanasam, Tamil Nadu, India, [email protected]

Abstract - The concept of M-fuzzy magic labeling is introduced. M- fuzzy magic labeling for some graphs like path, cycle and star graph are defined . It is proved that, Every M-fuzzy magic graph is a M-fuzzy labeling graph, but converse is not true .

Keywords: M-Fuzzy graph, M- Fuzzy Labeling, M- Fuzzy Magic Graph, Wheel graph, Fan graph.

1. INTRODUCTION: establish some result from the class of fuzzy graphs [1]. Graph Theory is a very important tool to represent many real world problems. In this paper, section 1.1 contains basic definitions and in section 1.2 a new concept of M- Nowadays, graphs do not represent all the system fuzzy magic labeling has been introduced. In properly due to the uncertainly of the parameters of section 2, we proved some proposition on M-fuzzy systems. Fuzzy is a newly emerging mathematical magic labeling for some graphs like path, cycle and frame work to exemplify the phenomenon of star. In section 3, we worked out some examples on uncertainty in real life tribulation. In 1965 Zadeh M-fuzzy magic labeling for some graphs wheel, introduced the concept of fuzzy by various fan, pan, helm, butterfly, and Petersen graphs. independent researches.viz; Rosenfeld [7], Yen and Bang [14] during 1970’s. Several fuzzy analogs of 1.1 PRELIMINARIES graph theoretic concepts such as path, cycle were explored by them. DEFINITION 1.1.1: A fuzzy graph G = (σ, µ) is a pair of function σ: V → [0, 1] and µ Though it is very young, it has been growing : V×V → [0, 1] , where ∀ u, v ∈ V we have , fast and has numbers application in various field. (u,v)  (u) (v) . Further research on fuzzy graphs has been witnessing an exponential growth both within DEFINITION 1.1.2: A graph G = (σ, µ) is mathematics and in its applications in science and said to be a fuzzy labeling graph if σ: V → [0,1] technology. and µ : V×V → [0 ,1 ] is bijective such that the membership value of edges and vertices are distinct In crisp graph, a bijection f: VUE → and N that assign to each vertex and/or edge if G = (V, E), a unique natural number is called a labeling. (u,v)  (u) (v) , ∀ u, v ∈ V. The concept of magic labeling in crisp graph was motivated by the notion of magic squares in DEFINITION 1.1.3: A fuzzy labeling graph is number theory. The notion of magic graph was first said to be a fuzzy magic graph if 휎(푢) + introduced by J. Sedlacek [8] in 1964. He µ(푢, 푣) + 휎(푣) has a same value for all 푢, 푣 ∈ 푉,

defined a graph to be magic if it has an edge which is denoted by m0(G). labeling, within the range of real numbers, such that the sum of the labels around any vertex equals some constant, independent of the choice of vertex. These labeling have been studied by B.M. Stewart [9] who called the labeling as supper magic if the labels are consecutive integers, starting from 1. Several others have studied these labeling. We www. jespublication.com Page No:1131 Vol 11, Issue 6,June/ 2020

ISSN NO: 0377-9254 EXAMPLE 1.2.2:

0.01 0.02 0.03

v1 v2 v3 v4 0.07 0.05 0.06 0.04

DEFINITION 1.1.4: A fuzzy graph G = (σ, µ) is called a M-fuzzy graph. If a pair of function σ: V → [0, 1] and µ: V×V → [0, 1] satisfied the DEFINITION 1.2.3: A M-fuzzy labeling graph is following conditions said to be a M- fuzzy magic graph if 휎(푢) + µ(푢, 푣) ≤ 휎(푢) or µ(푢, 푣) + 휎(푣) has a same value for all 푢, 푣 ∈ 푉, which is denoted by 푚(퐺). 휎(푢) ≤ µ(푢, 푣) ≤ 휎(푣), EXAMPLE 1.2.4: The above graph is true for M- then (u,v)  (u) (v) , ∀ u, v∈ V. fuzzy magic graph.

2. PROPERTIES OF M- FUZZY MAGIC DEFINITION 1.1.5: A path P in a fuzzy graph is a LABELING sequence of distinct nodes v1,v2 ,…, vn such that

μ(vi , vi+1) > 0 ; 1 ≤ 푖 ≤ 푛. PROPOSITION 2.1: For all 푛 ≥ 1, the path Pn is a M-fuzzy magic graph. DEFINITION 1.1.6: A path P is called a cycle if v1 = vn and 푛 ≥ 3. PROOF: Let P be any path with length n ≥ 1 and v v v and v v v v v v are the nodes DEFINITION 1.1.7: A star in a fuzzy graph 1, 2 , …, n 1 2 , 2 3 ,.., n-1 n and edges of P. Let z → (0, 1] such that one can consist of two node sets V and U with |V| = 1 and choose z = 0.1 if 푛 ≤ 4 and 푧 = 0.01 if |U| > 1, such that 휇(v, ui) > 0 and 휇(ui , ui+1) = 0 , 푛 ≥ 5. 1≤ i ≤ n. It is denoted by S1, n. If the length of the path P is odd then the M-fuzzy 1.2 M- FUZZY LABELING labeling is defined as follows: DEFINITION 1.2.1: A graph G = (σ, µ) 푛+1 σ(v2i-1) = (2n+2-i)z , 1 ≤ i ≤ is said to be a M- fuzzy labeling graph if σ: V → 2 [0,1] and µ : V×V → [0 ,1 ] is bijective 푛+1 such that the membership value of edges and σ(v2i) = Min{σ(v2i-1) / 1 ≤ i ≤ } – 2 vertices are distinct and satisfied the following 푛+1 conditions i(z) , 1 ≤ i ≤ 2 µ(푢, 푣) ≤ 휎(푢) or µ(vn-i+2, vn+1-i) = Max{σ(vi) / 1≤ i ≤ n+1} 휎(푢) ≤ µ(푢, 푣) ≤ 휎(푣), - Min{σ(vi) /1≤ i≤ n+1}

then (u,v)  (u) (v) ,∀푢, 푣 ∈ 푉. - (i-1)z , 1 ≤ i ≤ n

Case (i) : i is even

Then i = 2x for any positive integer x. y positive integer x

For each edge vi, vi+1

m(Pn) = σ(vi) + µ(vi , vi+1) + σ(vi+1)

= σ(v2x) + µ(v2x , v2x+1) + σ(v2x+1) www. jespublication.com Page No:1132 Vol 11, Issue 6,June/ 2020

ISSN NO: 0377-9254

푛+1 = Min{σ(v2i-1) / 1 ≤ i ≤ } – x(z) - (i- 1)z , 1 ≤ i ≤ n 2 Case (i) : i is even + Max {σ(vi) / 1≤ i ≤ n+1 } – Then i = 2x for any positive integer x Min{σ(vi) / 1≤ i ≤ n+1 } – For each edge v , v (n-2x)z + (2n-x+1)z i i+1

푛+1 m(Pn) = σ(vi) + µ(vi, vi+1) + σ(vi+1) = Min {σ(v2i-1) / 1 ≤ i ≤ } + 2 = σ(v2x) + µ(v2x , v2x+1) + σ(v2x+1) Max {σ(vi) / 1≤ i ≤ n+1 } - = (2n+2-x)z + Min {σ(vi) / 1≤ i ≤ n+1 } + Max {σ(vi) / 1≤ i ≤ n+1} (n+1)z - Min {σ(vi) / 1≤ i ≤ n+1} Case (ii) : i is odd – (n-2x)z + Then i = 2x +1 for any positive integer x 푛 Min { σ(v2i) / 1 ≤ i ≤ } 2 For each edge vi, vi+1 – (x+1)z m(Pn) = σ(vi) + µ(vi, vi+1) + σ(vi+1) 푛 = Min{σ(v2i) / 1 ≤ i ≤ } + =σ(v2x+1)+µ(v2x+1,v2x+2)+ σ(v2x+2) 2

= (2n-x + 1)z + Max { σ(vi) / 1≤ i ≤ n+1 } -

Max{σ(vi) / 1≤ i ≤ n+1 } - Min{σ(vi)/1≤i ≤ n+1 } + (n+1)z

Min{σ(vi) / 1≤ i ≤ n+1 } – Case (ii) : i is odd

(n-2x-1) + Then i = 2x +1 for any positive integer x

푛+1 For each edge vi, vi+1 Min{ σ(v2i-1) / 1 ≤ i ≤ } – 2

m(Pn) = σ(vi) + µ(vi , vi+1) + σ(vi+1) (x+1)z

푛+1 = σ(v2x+1)+µ(v2x+1,v2x+2) + σ(v2x+2) = Min{ σ(v2i-1) / 1 ≤ i ≤ } + 2 푛 = Min{σ(v2i) / 1 ≤ i ≤ } – (x+1)z – 2 Max{ σ(vi) / 1≤ i ≤ n+1 } - (n-2x-1)z + Min{σ(vi) / 1≤ i ≤ n+1 } + (n+1)z

Max {σ(vi) /1≤ i ≤ n+1 } - If the length of the path P is even, then the M-fuzzy

labeling is defined as follows: Min{σ(vi)/ 1≤ i ≤ n+1 } + (2n-x+1)z

푛 푛 σ (v2i) = (2n+2-i)z , 1 ≤ i ≤ =Min{ σ(v2i) / 1 ≤ i ≤ } + 2 2

푛 σ(v2i-1) = Min{σ(v2i) / 1 ≤ i ≤ } – Max{ σ(v i) / 1≤ i ≤ n+1 } - 2

푛+2 Min{σ(v ) / 1≤ i ≤ n+1 } + (n+1)z i(z) , 1 ≤ i ≤ i 2 Therefore in both cases the M-fuzzy magic value µ(vn-i+2, vn-i+1) = Max{σ(vi) /1≤ i≤ n+1} m(Pn) is same and unique. Thus Pn is M-fuzzy magic graph for all - Min{σ(vi) /1≤ i ≤ n+1} www. jespublication.com Page No:1133 Vol 11, Issue 6,June/ 2020

ISSN NO: 0377-9254

n ≥ 1. m(Cn) = σ(vi) + µ(vi , vi+1) + σ(vi+1)

PROPOSITION 2.2: If n is odd, then the cycle Cn =σ(v2x+1)+µ(v2x+1,v2x+2)+ σ(v2x+2) is a M-fuzzy magic graph. 푛−1 1 = Min { σ(v2i) / 1 ≤ i ≤ } – (x+1)z + 2 2 PROOF: Let Cn be any cycle with odd number of Max { σ(vi) / 1≤ i ≤ n } – (n-2x-1)z + (2n-x)z v1, v2 , …, vn and v1v2 , v2v3 ,..,vn-1vn are the nodes and

edges of of Cn . Let z → (0, 1] such that one can 1 = Max { σ(vi) / 1≤ i ≤ n } + Min { σ(v2i) / 1 2 choose z = 0.1 if n ≤ 3 and z = 0.01 if n ≥ 4. 푛−1 ≤ i ≤ } + n(z) 2 The M-fuzzy labeling for cycle is defined as follows: Therefore from above cases G is a M-fuzzy magic graph if it has odd number of nodes. 푛−1 σ(v2i) = (2n+1-i)z , 1 ≤ i ≤ 2 PROPOSITION 2.3: For any n ≥ 2, star S1,n is a 푛−1 M-fuzzy magic graph. σ(v2i-1) = Min { σ(v2i) / 1 ≤ i ≤ } 2 PROOF: Let S be a star graph with v,u u u 푛+1 1,n 1 , 2 , …, n – i(z) , 1 ≤ i ≤ 2 as nodes and vu1 , vu2 ,..,vun as edges.

1 µ(v1 , vn) = Max { σ(vi) / 1≤ i ≤ n } Let z → (0, 1] such that one can choose z = 0.1 if n 2 ≤ 4 and z = 0.01 if n ≥ 5.

µ(vn-i+1 , vn-i) = µ(v1 , vn) – i(z) , The M-fuzzy labeling for star graph is defined as 1≤ i ≤ n-1 follows:

Case (i) : i is even σ(ui) = [2(n+1)-i]z , 1 ≤ i ≤ n

Then i = 2x for any positive integer x σ(v) = Min { σ(ui) / 1 ≤ i ≤ n } – z

For each edge vi, vi+1 µ(v, un-i) =Max{σ(ui), σ(v)/1≤ i ≤ n}

m(Cn) = σ(vi) + µ(vi, vi+1) + σ(vi+1) -Min{σ(ui), σ(v)/1≤ i ≤ n}

=σ(v2x)+µ(v2x,v2x+1)+ σ(v2x+1) - i(z) , 0 ≤ i ≤ n-1.

= (2n+1-x)z + Case (i) : i is even

1 Then i = 2x for any positive integer x Max { σ(vi) / 1≤ i ≤ n } – 2

For each edge v, ui (n-2x)z +

푛−1 m(S1,n) = σ(v) + µ(v, ui) + σ(ui) Min{σ(v2i) / 1 ≤ i ≤ } 2 = σ(v) + µ(v, u2x) + σ(u2x) – (x+1)z = Min {σ(ui) / 1≤ i ≤ n } - z + 1 = Max { σ(vi) / 1≤ i ≤ n } + 2 Max { σ(ui) , σ(v) / 1≤ i ≤ n } 푛−1 Min{σ(v2i) /1 ≤ i ≤ }+ n(z) 2 - Min{σ(ui), σ(v) / 1≤ i ≤ n } - (n-2x)z + [2(n+1) – 2x]z Case (ii) : i is odd

= Min {σ(ui) / 1≤ i ≤ n } + Then i = 2x +1 for any positive integer x

Max {σ(ui) , σ(v) / 1≤ i ≤ n } – For each edge vi, vi+1

Min { σ(ui) , σ(v) / 1≤ i ≤ n } www. jespublication.com Page No:1134 Vol 11, Issue 6,June/ 2020

ISSN NO: 0377-9254 + (n+1)z SOME EXAMPLE OF M-FUZZY MAGIC GRAPH Case (ii) : i is odd BUTTERFLY GRAPH: The butterfly graph is a Then i = 2x + 1 for any positive integer x planar undirected graph with 5 vertices and 6 edges. For each edge v, ui EXAMPLE 3.1: m(S1,n) = σ(v) + µ(v, ui) + σ(ui)

= σ(v) + µ(v , u2x+1) + σ(u2x+1)

= Min {σ(ui) / 1≤ i ≤ n } - z + v1 0.25 Max {σ(ui) , σ(v) / 1≤ i ≤ n }

– Min{σ(ui) , σ(v) / 1≤ i ≤ n }

– (n-2x-1)z + [2(n-x)]z

= Min {σ(ui) / 1≤ i ≤ n } + PAN GRAPH: The pan graph is the graph obtained by joining a cycle graph to a singleton Max{σ(u ) , σ(v) / 1≤ i ≤ n } i graph with a bridge. The special case of the 3-pan graph is sometimes known as the paw graph and – Min{σ(ui) , σ(v) / 1≤ i ≤ n } the 4-pan graph as the banner graph. + (n+1)z EXAMPLE 3.3: Therefore from above cases G is a M-fuzzy magic graph.

PROPOSITION 2.4: Every M-fuzzy magic graph is a M-fuzzy labeling graph.

PROOF: This is immediate from the definition (M-fuzzy magic graph)

REMARK 2.5: Converse of the above theorem is not true.

BULL GRAPH: The bull graph is a planar undirected graph with 5 vertices and 5 edges, in the form of a triangle with two disjoint pendent edges.

www. jespublication.com Page No:1135 Vol 11, Issue 6,June/ 2020

ISSN NO: 0377-9254 EXAMPLE 3.3: HELM GRAPH: The helm graph is the graph obtained from a wheel graph by adjoining a pendent edge at each vertex of the cycle. EXAMPLE 3.6:

WHEEL GRAPH:

A wheel graph is a graph formed by connecting a single vertex to all the vertices of a cycle. For n ≥ 4, the wheel Wn is defined to be the graph K1 + Cn- 1. PETERSEN GRAPH: The Petersen Graph is a EXAMPLE 3.4: non planar graph with 10 vertices and 15 edges.

EXAMPLE 3.7:

FAN GRAPH: A fan graph Fm,n is defined as the graph join Km + Pn , where Km is the empty graph REFERENCE: on m vertices and Pn is the path graph on n vertices . The case m = 1 corresponds to the usual fan [1] Jahir Hussain. R and Kathikkeyan R.M., Fuzzy graphs, while m = 2, corresponds to the double fan, Graceful Graphs, Advances in Fuzzy Mathematics. etc… ISSN 0973 – 533X Volume 12, Number 2 (2017), EXAMPLE 3.5: pp.309-317. Research India Publications. [2] Jamil R.N., Javaid M., ur Rehman M.A., Kirmani K.N., On the construction of fuzzy magic graphs, Science International 28(3)(2016).

[3] Nagoorgani. A and Chandrasekaran V.T., A First Look at Fuzzy Graph Theory, Allied Publishers Pvt. Ltd, 2010.

[4] Nagoorgani A. and Rajalaxmi(a) Subahashini D., Propertics of Fuzzy Labeling Graph , Applied Mathematical Science, vol. 6,no. 69-72,pp. 3461- 3466,2012. www. jespublication.com Page No:1136 Vol 11, Issue 6,June/ 2020

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Mathematical Archive, Vol. 4, No. 2, 2014, 88-95.

[6] Nagoorgani A., Muhammad Akram and Rajalaxmi(a) Subahashini D., Novel Propertics of Fuzzy Labeling Graphs, Journal of Mathematics, Volume 2014.

[7] Rosenfeld .A, Fuzzy Graph, In: Zadeh L.A., Fu K.S. and Shimura M., Editors, Fuzzy Sets and their

Applications to Cognitive and Decision Process, Academic Press, New York (1975) 77-95.

[8]Sedlacek J., Theory of graphs and its applcations, Smolenice Symposium (Prague,1964),

163 – 164, problem 27.

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R.S., Fuzzy Magic Graphs – A Brief Study, International Journal of Pure and Applied Mathematics, Volume 119 No. 15, 2018, 1161 – 1170. [11] Trenkler M., Some results on magic graphs, in “Graphs and other combinatorial topics” Proc. Of the 3rd Czechoskovak Symp., Prague , 1983, edited

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[14] Yeh R.T. and Bang S.Y., Fuzzy relations, fuzzy graphs and their applications to clustering analysis. In : L.A. Zadeh, K..S. Fu and M.Shirmura, Editors, Fuzzy Sets and Their Applications, Academic Press (1975), 125 – 149.