The degree of vertices in fuzzy graphs obtained from

DEGREE OF FUZZY GRAPH *A. Nagoor Gani and **K. Radha P.G. & Research Department of Mathematics, Jamal Mohamed College(Autonomous), Trichy–20, India. P.G. Department of Mathematics, Arignar Anna Government Arts College, Musiri – 621201, India.

*e-mail: [email protected] **e-mail: [email protected] ABSTRACT A fuzzy graph can be obtained from two given fuzzy graphs using cartesian product, composition, union and join. In this paper, we find the degree of a vertex in fuzzy graphs formed by these operations in terms of the degree of vertices in the given fuzzy graphs. Key Words: Degree of a vertex, Regular fuzzy graph, cartesian product, composition, union and join. 1.Introduction Fuzzy graph theory introduced by Azriel Rosenfeld in 1975 has been growing fast and has numerous applications in various fields. The operations of cartesian product, composition, union and join on two fuzzy graphs were defined by Mordeson.J.N., and Peng.C.S. In this paper, we study about the degree of a vertex in fuzzy graphs, which are obtained from two given fuzzy graphs using these operations. If the fuzzy graph G is formed from two fuzzy graphs G1 and G2, we find the degree of vertices in cartesian product, composition, union and join of G1 and G2 in terms of the degree of vertices of G1 and G2 under some restrictions. First we go through some basic definitions, which can be found in [1]-[5]. 2. Basic Definitions Throughout this paper, we shall denote the edge between two vertices u and v by uv instead of (u,v) because in cartesian product or composition of two fuzzy graphs, a vertex is an ordered pair. 2.1 Fuzzy graph A fuzzy graph G is a pair of functions G:(σ,μ) where σ is a fuzzysubset of a non empty set V and μ is a symmetric fuzzy relation on σ, (i.e.) μ(uv) ≤ σ (u)∧ σ (v). The underlying crisp graph of G:(σ,μ) is denoted by G*:(V,E) where E  VxV. 2.2 Order of a Fuzzy graph The order of a fuzzy graph G is defined by O(G) =  σ (u) where the summation is over all u∈E. 2.3. Degree of a vertex Let G:(σ,μ) be a fuzzy graph. The degree of a vertex u is defined by dG(u)=μ(uv) = μ(uv). u≠v

uv∈E

Note: Throughout this paper G1:(σ1,μ1) and G2:(σ2,μ2) denote two fuzzy graphs with underlying crisp graphs G1*:(V1,E1) and G2*:(V2,E2). Also let us take | Vi | =pi, i = 1,2.

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2.4. Cartesian product The cartesian product of two fuzzy graphs G1 and G2 is defined as a fuzzy graph G = G1 x G2 :( σ1x σ2 , μ1 x μ2 ) on G*:(V,E) where V=V1 x V2 and E={((u1,u2)(v1,v2)) / u1=v1 , u2v2∈E2 or u2=v2 ,u1v1∈E1} ( σ1x σ2 )(u1,u2) = σ1 (u1)∧ σ2 (u2),

with and

for all (u1,u2) ∈V1 x V2

(μ1 x μ2 )((u1,u2)(v1,v2)) = σ1 (u1) ∧μ2 (u2v2),

if u1=v1 , u2v2∈E2

= σ2 (u2) ∧ μ1(u1v1),

if u2=v2 , u1v1∈E1

2.5. Composition The Composition of two fuzzy graphs G1 and G2 is defined as a fuzzy graph G= G1 [ G2 ]:( σ1 ∘ σ2 , μ1 ∘ μ2 ) on G*:(V,E) where V= V1 x V2 and E={((u1,u2)(v1,v2)) / u1=v1 , u2v2∈E2 or u2=v2 ,u1v1∈E1 or u2≠v2 ,u1v1∈E1} (σ1∘ σ2 )(u1,u2) = σ1 (u1)∧ σ2 (u2),

with

for all (u1,u2) ∈V1 x V2

and (μ1 ∘ μ2 )((u1,u2),(v1,v2)) = σ1 (u1) ∧ μ2 (u2v2),

if u1=v1 , (u2,v2)∈E2

= σ2 (u2) ∧ μ1(u1v1),

if u2=v2 ,( u1,v1)∈E1

= σ2 (u2) ∧σ2 (v2) ∧ μ1(u1v1),

if u2≠v2 ,(u1,v1)∈E1

2.6 Union The union of two fuzzy graphs G1 and G2 is defined as a fuzzy graph G=G1G2:( σ1  σ2 , μ1  μ2 ) on G*:(V,E) where V= V1  V2 and E=E1  E2 , with

and

(σ1  σ2 )(u) = σ1 (u),

if u ∈V1 - V2

= σ2 (u),

if u∈V2 - V1

= σ1 (u) ⋁ σ2 (u) ,

if u∈V1 ∩ V2

(μ1  μ2 )(uv) = μ1(uv),

if uv ∈E1 –E2

= μ2(uv),

if uv ∈E2 –E1

=μ1(uv)⋁ μ2(uv),

if uv ∈E1 ∩E2

2.7 Join of two fuzzy graphs Assume that V1 ∩ V2 = . The join (sum) of G1 and G2 is defined as a fuzzy graph G= G1 + G2 :( σ1 + σ2 , μ1 + μ2 ) on G*:(V,E) where V= V1V2 and E=E1E2E’ where E’ is the set of all edges joining vertices of V1 with vertices of V2, with

(σ1 + σ2 )(u) = (σ1  σ2 )(u)

for all u ∈V1  V2

and

(μ1 + μ2 )(uv) = (μ1  μ2 )(uv),

if uv ∈ E1  E2

= σ1 (u) ⋁ σ2 (u) , if uv ∈ E’ In the following sections, we find the degree of vertices in cartesian product, composition, union and join of G1 and G2 in terms of the degree of vertices of G1 and G2 under some conditions.

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3. Degree of a vertex in Cartesian product 3.1 Theorem: Let G1:(σ1,μ1) and G2:(σ2,μ2) be two fuzzy graphs. If σ1 ≥ μ2 and σ2 ≥ μ1, then d G1 x G2(u1,u2) = d G1(u1) + d G2 (u2). Proof: By definition, for any (u1,u2) ∈ V1 xV2, d G1 x G2(u1,u2) = ∑ (μ1 x μ2 )((u1,u2)(v1,v2)) (u1,u2)(v1,v2)∈E

= ∑ σ1 (u1) ∧ μ2 (u2v2) + ∑ σ2 (u2) ∧ μ1(u1v1) u1=v1,u2v2∈E2

= ∑ μ2 (u2v2)

u2=v2,u1v1∈E1

+ ∑ μ1(u1v1) , ( since σ1 ≥ μ2 and σ2 ≥ μ1)

u2v2∈E2

u1v1∈E1

= dG2(u2) + dG1(u1) 3.2 Example: Consider the fuzzy graphs G1:(1, 1) and G2:(2, 2) in Fig.1. G1 x G2 G2 G1 (u1, u2) (u1, v2) u (. 3 ) 2 u1(.2) .1 (.2) (.2) .2

.1

v1(.5)

v2(.4)

.2 (.3) (v1, u2)

.2 .1

(.4) (v1, v2)

Fig. 1 Here 2  2 and 2  1. So by theorem 3.1, d G1 x G2 (u1, u2) = d G2 (u2) + dG1(u1) = 0.2 + 0.1 = 0.3 This can be verified in the figure of G1 x G2 3.3. Theorem: Let G1:(σ1,μ1) and G2:(σ2,μ2) be two fuzzy graphs and let σi(u)=ci for all u∈ Vi , where ci is a constant, i=1,2. (i). If σ1 ≤ μ2, then d G1 x G2(u1,u2) = d G1(u1) + c1d G2* (u2). (ii). If σ2 ≤ μ1, then d G1 x G2(u1,u2) = d G2(u2) + c2d G1* (u1). where d Gi* (ui) is the degree of ui in Gi* Proof: Since σ1 and σ2 are constants, μ1 ≤ σ1 and μ2 ≤ σ2. (i). We have σ1 ≤ μ2. Hence μ1 ≤ σ1 ≤ μ2 ≤ σ2. So σ2 ≥ μ1. Now by definition, for any (u1,u2) ∈ V1 xV2, d G1 x G2(u1,u2) = ∑ (μ1 x μ2 )((u1,u2)(v1,v2)) (u1,u2)(v1,v2)∈E

= ∑ σ1 (u1) ∧ μ2 (u2v2) + ∑ σ2 (u2) ∧ μ1(u1v1) u1=v1,u2v2∈E2

u2=v2,u1v1∈E1

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= ∑ σ1 (u1) + ∑ μ1(u1v1) , ( since σ1 ≤ μ2 and σ2 ≥ μ1) u2v2∈E2

= ∑ c1

u1v1∈E1

+ dG1(u1)

u2v2∈E2

= c1 dG2*(u2) + dG1(u1) (ii). Proof is similar to (i). 3.4 Example: Consider the fuzzy graphs G1:(1, 1) and G2:(2, 2) in Fig.2. G1 x G2 G2 G1 .3 (u1,u2) (u1, v2) u2(.5) u1(.3) .3 .2 .2 .3 .2 .4 .2 (v1 , w2) v1(.3)

w2 (.5)

(.5) v2

.3 (v1 , v2)

.3

(v1 , u2)

Fig.2 Here 1 and 2 are constants with 1 2. So by theorem 3.3, d G1 x G2 (u1, u2) = c1d G2* (u2) + d G1 (u1) = 0.3 x 2 + 0.2 = 0.6 + 0.2 = 0.8 This can be verified in the figure of G1 x G2. In G1 x G2 , (1x 2) (u,v) = 0.3 for all (u,v) V1x V2 4. Degree of a vertex in Composition 4.1 Theorem: Let G1:(σ1,μ1) and G2:(σ2,μ2) be two fuzzy graphs. If σ1 ≥ μ2 and σ2 ≥ μ1, then d G1[ G2](u1,u2) = p2d G1(u1) + d G2 (u2). Proof: By definition, for any (u1,u2) ∈ V1xV2, d G1[ G2](u1,u2) = ∑ (μ1 ∘ μ2 )((u1,u2)(v1,v2)) (u1,u2)(v1,v2)∈E

= ∑ σ1 (u1) ∧ μ2 (u2 v2)

+ ∑ σ2 (u2) ∧ μ1(u1 v1)

u1=v1 , u2v2∈E2

u2=v2 ,u1v1∈E1

+ ∑ σ2 (u2) ∧ μ1(u1 v1) u2≠v2 ,u1v1∈E1

= ∑ μ2 (u2,v2) u2v2∈E2

+ ∑ μ1(u1 v1) + ∑ μ1(u1 v1) u1v1∈E1 u2=v2

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u1v1∈E1 u2≠v2

( since σ1 ≥ μ2 and σ2 ≥ μ1)

(u1,w2)

= dG2(u2) + | V2 | ∑ μ1(u1 v1) u1v1∈E1

= dG2(u2) + p2 dG1(u1) 4.2 Example : Consider the fuzzy graphs G1 and G2 in Example 3.2. By theorem 4.1, d G1 [G2] (u1, u2) = d G2 (u2) + p2 dG1(u1) = 0.1 + 2 x 0.2 = 0.1 + 0.4 = 0.5 This can be verified in the figure of G1[G2] given in Fig.3 G1 x G2 (u1, u2) (u1, v2) ..1 (.2) (.2) .2 .2 .2 .2 (.3) (v1, u2)

.1

(.3) (v1, v2)

Fig. 3 4.3. Theorem: Let G1:(σ1,μ1) and G2:(σ2,μ2) be two fuzzy graphs and let σi(u)=ci for all u∈ Vi , where ci is a constant, i=1,2. (i). If σ1 ≤ μ2, then d G1[G2](u1,u2) = p2d G1(u1) + c1d G2* (u2). (ii). If σ2 ≤ μ1, then d G1[G2](u1,u2) = d G2(u2) + p2c2d G1* (u1). Proof: Since σ1 and σ2 are constants, μ1 ≤ σ1 and μ2 ≤ σ2. (i).We have σ1 ≤ μ2. Hence μ1 ≤ σ1 ≤ μ2 ≤ σ2. So σ2 ≥ μ1. By definition, for any (u1,u2) ∈ V1xV2, d G1[ G2](u1,u2) = ∑ (μ1 ∘ μ2 )((u1,u2)(v1,v2)) (u1,u2)(v1,v2)∈E

= ∑ σ1 (u1) ∧ μ2 (u2v2)

+ ∑ σ2 (u2) ∧ μ1(u1v1)

u1=v1 , u2v2∈E2

u2=v2 ,u1v1∈E1

+ ∑ σ2 (u2) ∧ μ1(u1v1) u2≠v2 ,u1v1∈E1

= ∑ σ1 (u1) u2v2∈E2

+ ∑ μ1(u1v1) + ∑ μ1(u1v1) u1v1∈E1 u2=v2

u1v1∈E1 u2≠v2

= ∑ c1 + | V2 | ∑ μ1(u1v1) u2v2∈E2

( since σ1≤ μ2 and σ2 ≥ μ1)

u1v1∈E1

= c1 dG2*(u2) + p2 dG1(u1) (ii). We have σ2 ≤ μ1. Hence μ2 ≤ σ2 ≤ μ1 ≤ σ1. So σ1 ≥ μ2. By definition, for any (u1,u2) ∈ V1xV2,

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d G1[ G2](u1,u2) = ∑ (μ1 ∘ μ2 )((u1,u2)(v1,v2)) (u1,u2)(v1,v2)∈E

= ∑ σ1 (u1) ∧ μ2 (u2v2)

+ ∑ σ2 (u2) ∧ μ1(u1 v1)

u1=v1 , u2v2∈E2

u2=v2 ,u1v1∈E1

+ ∑ σ2 (u2) ∧ μ1(u1v1) u2≠v2 ,u1v1∈E1

= ∑ μ2 (u2v2)

+ ∑ σ2 (u2) + ∑ σ2 (u2)

u2v2∈E2

u1v1∈E1 u2=v2

u1v1∈E1 u2≠v2

( since σ1≥ μ2 and σ2≤ μ1)

= dG2(u2) + | V2 | ∑ c2 u1v1∈E1

= dG2(u2) + p2c2dG1*(u1) 4.4 Example: Consider the fuzzy graphs G1 and G2 in Fig. 4. G1 u1(.3)

G2 u2(.3)

.1

.4

v1(.3)

v2(.5)

(u1, u2)

G1 [G2] .3

(.3) .1 (.3) (v1, u2)

.1

.1

(u1, v2) (.3) .1

.3

(.3) (v1, v2)

Fig.4 Here 1 and 2 are constant with 1 2. So by theorem 4.3, d G1 [G2] (u1, u2) = c1d G2* (u2) + p2 dG1(u1) = 0.3 + 1 x 2 x 0.1 = 0.3 + 0.2 = 0.5 This can be verified in the figure of G1[G2]. 5. Degree of a vertex in union For any u ∈ V1  V2, we have three cases to consider. Case 1: Either u ∈ V1 or u ∈ V2 but not both. Then no edge incident at u lies in E1 ∩ E2. So (μ1  μ2 )(uv) = μ1(uv), = μ2(uv),

if u ∈ V1 (i.e.) if uv ∈ E1 if u ∈ V2 (i.e.) if uv ∈E2

Hence if u ∈ V1, then d G1G2(u) = ∑ μ1(uv) = dG1(u) uv∈E1

if u ∈ V2, then d G1G2(u) = ∑ μ2(uv) = dG2(u) uv∈E2

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Case 2: u ∈ V1 ∩ V2 but no edge incident at u lies in E1 ∩ E2. Then any edge incident at u is either in E1 or in E2 but not both. Also all these edges will be included in G1G2. Hence d G1G2(u) = ∑(μ1  μ2 )(uv) uv∈E = ∑ μ1(uv) + ∑ μ2(uv) uv∈E1

uv∈E2

= dG1(u) + dG2(u) Case 3: u ∈ V1 ∩ V2 and some edges incident at u are in E1 ∩ E2. Any edge uv which is in E1 ∩ E2 appear only once in G1G2 and for this uv, μ(uv)=μ1(uv)⋁ μ2(uv) By definition, d G1G2(u) = ∑(μ1  μ2 )(uv) uv∈E =

∑ μ1(uv) + ∑ μ2(uv) + ∑ μ1(uv)⋁ μ2(uv) uv∈E1-E2

uv∈E2-E1

uv∈E1 ∩ E2

= [ ∑ μ1(uv) + ∑ μ2(uv) + ∑ μ1(uv)⋁ μ2(uv) + ∑ μ1(uv)⋀ μ2(uv) ] uv∈E1-E2

uv∈E2-E1

uv∈E1 ∩ E2

uv∈E1 ∩ E2

- ∑ μ1(uv) ⋀ μ2(uv) uv∈E1 ∩ E2

=

∑ μ1(uv) + ∑ μ2(uv) - ∑ μ1(uv)⋀μ2(uv) uv∈E1

uv∈E2

uv∈E1 ∩ E2

= dG1(u) + dG2(u) - ∑ μ1(uv)⋀μ2(uv) uv∈E1 ∩ E2

5.1 Example : Consider the fuzzy graphs G1 : (1, 2) and G2 : (2, 2) in Fig.5. G2 G1 G2 G 1

(.5) u

.4

.3 (.9) w

v (1)

u (.7)

v (.5)

.5

.4

.3

(.9) w

w (.4)

Fig. 5 Consider x : Here x  V1. so by case 1, d G1 G2 (x) = d G1(x) = 0.5 + 0.6 = 1.1 Consider v: We have v  V1V2 but no edge incident at v lies in E1E2. So by case 2, dG1 G2 (v) = d G1(v) + d G1(v) = 1 + 0.3 = 1.3 Consider u: We have u  V1V2 and u w  E1E2. So by case 3, d G1 G2 (u) = dG1(u) + dG2(u) - 1 (uw) 2 (uw)

7

v (1)

.4

.4

.6 x (.6)

(.7) u

.3 .5

.6 x (.6)

= 0.7 + 0.4 – 0.3 = 0.8 All these degrees can be verified in the figure of G1 G2. 6. Degree of a vertex in Join Here V1 ∩ V2 =φ. So E1 ∩ E2 =. So

if uv ∈ E1

(μ1  μ2 )(uv) = μ1(uv),

if uv ∈E2

= μ2(uv),

d G1+ G2 (u) = ∑ μ1 (u,v) + ∑ σ1 (u) ⋁ σ2 (v) uv∈E1E2

uv∈E’

For any u ∈ V1, d G1+ G2 (u) = ∑ μ1 (u,v) + ∑ σ1 (u) ⋁ σ2 (v) uv∈E1

uv∈E’

= dG1(u) + ∑ σ1 (u) ⋁ σ2 (v)

(1)

v∈V2

Similarly, for any u ∈ V2, d G1+ G2 (u) = dG2(u) + ∑ σ1 (v) ⋁ σ2 (u)

(2)

v∈V1

6.1. Theorem: Let G1:(σ1,μ1) and G2:(σ2,μ2) be two fuzzy graphs. (i). If σ1 ≥ σ2, then d G1+G2(u1,u2) = dG1(u) + O (G2),

if u ∈ V1

= dG2(u) + p1 σ2(u),

if u ∈ V2

(ii). If σ2 ≥ σ1, then d G1+G2(u1,u2)

= dG1(u) + p2 σ1(u), = dG2(u) + O (G1),

Proof: (i). We have σ1 ≥ σ2. From (1), for any u ∈ V1, d G1+ G2 (u) = dG1(u) + ∑ σ1 (u) ⋁ σ2 (v) v∈V2

= dG1(u) + ∑ σ2 (v) v∈V2

= dG1(u) + O(G2) From (2), for any u ∈ V2, d G1+ G2 (u) = dG2(u) + ∑ σ1 (v) ⋁ σ2 (u) v∈V1

= dG2(u) + ∑ σ2 (u) v∈V1

= dG2(u) + p1 σ2 (u)

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if u ∈ V1 if u ∈ V2

(ii) Proof is similar. 6.2 Example : Consider G1 and G2 in Example 3.2. we have 2  2. So by (2) of theorem 6.1, dG1+G2 (u1) = dG1 (u1) + p2 2 (u1) = 0.2 + 2 x 0.2 = 0.2 + 0.4 = 0.6 dG1+G2 (u2) = dG2 (u2) + O (G1) = 0.1 + 0.5 = 0.6 These degrees can be verified in the figure of G1 + G2 given in fig. 6. G1+G2 (.2) u1

.2 .2

.3

.2 (.3) v1

u2 (.3)

.3 Fig.6

.3 .1 v2 (.6)

7. Conclusion In this paper, we have found the degree of vertices in G1x G2 , G1[G2] ,G1G2, and G1+ G2 in terms of the degree of vertices in G1 and G2 under some conditions and illustrated them through examples. So in such cases, knowing the properties of G1 and G2 will be enough to find the degree of vertices in Cartesian product, composition, union and join. They will be helpful especially when the graphs are very large. Also they will be useful in studying various properties of Cartesian product, composition, union and join of two fuzzy graphs. References [1] P.Bhattacharya, Some remarks on fuzzy graphs, Pattern recognition Lett. 6 (1987) 297-302. [2] John N.Modeson and Premchand S.Nair, Fuzzy Graphs and Fuzzy Hypergraphs, Physica-verlag Heidelberg 2000. [3] Mordeson, J.N. and Peng, C. S., Operations on Fuzzy Graphs, Inform. Sci. 79: 159-170, 1994. [4] A.Nagoor Gani and M.Basheer Ahamed, Order and Size in Fuzzy Graph, Bulletin of Pure and applied sciences. Vol.22E (No.1)2003, 145-148. [5]

A. Rosenfeld, Fuzzy Graphs, In: L. A. Zadeh, K.S. Fu, M. Shimura, Eds., Fuzzy sets and Their Applications, Academic Press (1975), 77-95.

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