degree, order and size in intuitionistic fuzzy graphs

International Journal of Algorithms, Computing and Mathematics Volume 3, Number 3, August 2010 ©Eashwar Publications

Degree, Order and Size in Intuitionistic Fuzzy Graphs A.Nagoor Gani P .G and Research Department of Mathematics, Jamal Mohamed College, Tiruchirappalli 620 020, India [email protected]

S. Shajitha Begum P .G and Research Department of Mathematics, Jamal Mohamed College, Tiruchirappalli 620 020, India [email protected] Abstract In this paper, we examine the properties of various types of degrees, order and size of intuitionistic fuzzy graphs and a new definition for complete intuitionistic fuzzy graph and intuitionistic regular fuzzy graph is given.

Keywords: Intuitionistic fuzzy graph, Effective Degree, Neighbourhood Degree, Order, Size 2010 Mathematics Subject Classification : 03E72, 03F55 1. Introduction Atanassov [1] introduced the concept of intuitionistic fuzzy (IF) relations and intuitionistic fuzzy graphs (IFGs). Research on the theory of intuitionistic fuzzy sets (IFSs) has been witnessing an exponential growth in Mathematics and its applications. This ranges from traditional Mathematics to Information Sciences. This leads to consider IFGs and their applications. R. Parvathy and M.G.Karunambigai’s paper [2] introduced the concept of IFG and analyzed its components. In this paper, we discuss various types of degree and some properties. 2. Preliminaries 2.1. Definition: An Intuitionistic fuzzy graph is of the form G = < V, E > where (i) V={v1,v2,….,vn} such that µ1: V [0,1] and γ1: V [0,1] denote the degree of membership and nonmembership of the element vi V, respectively, and 0 ≤ μ1 (vi) + γ1 (vi) ≤ 1 for every vi V, (i = 1,2, ……. n), (1) (ii) E V x V where µ2: VxV [0,1] and γ2: VxV [0,1] are such that μ2 (vi , vj) ≤ min [μ1 (vi), μ1 (vj)], (2) γ2 (vi , vj) ≤ max [γ1 (vi), γ1 (vj) ] (3) and 0 ≤ μ2 (vi, vj) + γ2 (vi,vj) ≤ 1 for every (vi vj) E, ( i, j = 1,2, ……. n) (4)

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International Journal of Algorithms, Computing and Mathematics

2.2. Definition: An IFG H = < V’, E’ > is said to be an Intuitionistic fuzzy subgraph (IFSG) of the IFG, G = < V, E > if V’ V and E’ E. In other words, if µ1i’ ≤ µ1i ; γ1i’ ≥ γ1i and µ2ij’ ≤ µ2ij ; γ2ij’ ≥ γ2ij for every i, j = 1,2………n. (5) (0.7,0.2) V2(0.7,0.2)

V1(0.8,0.1) (0.5,0.1)

(0.6,0.3) V1(0.7,0.2)

V2(0.6,0.3)

(0.5,0.3) (0.5,0.3)

100

(0.5,0.3) (0.7,0.2)

(0.6,0.1) 80

V4(0.9,0.1)

60

V3(0.8,0.2)

V4(0.8,0.2)

(0.8,0.2)

(0.5,0.3) (0.6,0.3) East

V3(0.5,0.3)

West

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Fig. 2. Complete North Intuitionistic fuzzy graph

Fig. 1 Intuitionistic fuzzy graph 20

2.3. Definition: An Intuitionistic fuzzy graph is complete if 0 1st Qtr 2nd Qtr 4th Qtr µ2ij = min ( µ1i, µ1j ) and γ2ij = maxQtr (γ 3rd 2i ,γ 2j) for all ( vi , vj )

V.

(6)

3. Degree of a Vertex 3.1. Definition: Let G = < V,E > be an IFG. Then the degree of a vertex v is defined by d(v) = (dμ(v), dγ(v)) where dμ(v) = Σ u≠v μ2(v,u) and dγ(v) = Σ u≠v γ2(v,u). 3.2. Definition: The minimum degree of G is δ(G) = (δμ(G), δγ(G)) where δμ (G) = Λ {dμ (v)/v V} and δγ (G) = Λ {dγ (v)/v V} 3.3. Definition: The maximum degree of G is Δ (G) = (Δμ (G), Δγ (G)) where Δμ (G) = V{dμ (v)/v V} and Δγ (G) = V {dγ (v) / v V} (0.4,0.4) V2(1,0)

V1(0.5,0.4) (0.2,0.6) (0.2,0.5)

(0.2,0.7)

V4(0.3,0.5)

V3(0.2,0.7) (0.2,0.6)

Fig. 3. Intuitionistic fuzzy graph

3.4. Example: In fig 3.

d(v1)=(0.8,1.5)

d(v2)=(0.6,1.1)

d(v3)=(0.6,1.9)

d(v4)=(0.4,1.1)

δ (G) = (0.4 , 1.1)

Δ (G) = (0.8, 1.9)

3.5. Proposition: The sum of the degree of membership value of all vertices in an IFG is equal to twice the sum of the membership value of all edges and the sum of the degree of nonmembership value of all vertices in an IFG is equal to twice the sum of the nonmembership value of all edges. (i.e.) Σ d(vi) = [ Σ dμ(vi) , Σ γ(vi)] = [ 2 Σ v≠u μ2(v,u), 2 Σ v≠u γ2(v,u)] (7) Proof: Let G = < V, E > be an IFG where V={v1,v2,……..vn} Σd(vi) =

[ Σ dμ(vi), Σdγ(vi)] 12

Degree, Order and Size in Intuitionistic Fuzzy Graphs

=

[dμ(v1),dγ(v1) + dμ(v2),dγ(v2) + ……….. + dμ(vn),dγ(vn)]

=

[μ2(v1,v2),γ2(v1,v2)+μ2(v1,v3),γ2(v1,v3)+….+μ2(v1,vn),γ2(v1,vn) + μ2 (v2,v1),γ2(v2,v1)+ μ2 (v2,v3), γ2(v2,v3)+………+ μ2 (v2,vn),γ2(v2,vn) + ……………………………………………………………………. + μ2(vn,v1),γ2(vn,v1)+ μ2 (vn,v2),γ2(vn,v2)+…+ μ2 (vn-1,vn),γ2(vn-1,vn)]

=

2[μ2(v1,v2),γ2(v1,v2)+μ2(v1,v3),γ2(v1,v3)+….+μ2(v1,vn),γ2(v1,vn)]

=

[2Σ μ2 (v, u), 2Σγ2 (v, u)].

Hence the proof 3.6. Proposition: The number of vertices of odd membership degree and the number of vertices of odd nonmembership degree in an IFG is even. Proof: Let G = < V, E > be an IFG. If we consider the vertices with odd and even membership degrees respectively, then Σ dµ (vi) = Σodd dµ (vj) + Σeven dµ (vk) (8) where Σ dµ (vi) is the sum of the membership degrees of all vertices in G which can be expressed as the sum of 2 sums each taken over vertices of even and odd membership degrees respectively. Since Σ dµ (vi) is even by equation (7) and Σeven dµ (vk) is even being a sum of even numbers, Σodd dµ (vj) is also even. i.e. Σodd dµ (vj) = a even number. Since each dµ (vj) is odd, the total number of terms in the sum must be even to make the sum an even number. Similarly, we can prove Σodd dγ (vj), the number of vertices of odd nonmembership degree is even. 3.7. Example: V1(0.1,0.1)

(0.1,0.1)

(0.1,0.2)

V3(0.5,0.4)

V2(0.1,0.2) (0.2,0.1)

Fig. 4. Intuitionistic fuzzy graph

d(v1) = (0.2,0.3)

d(v2) = (0.3,0.3)

d(v3) = (0.3,0.2)

3.8. Proposition: The maximum degree of any vertex in an IFG with n vertices is n-1. Proof: Let G = < V, E > be an IFG. The maximum membership value given to an edge is 1 and the number of edges incident on a vertex can be at most n-1. Hence the maximum membership degree dµ (vi) of any vertex vi in an IFG with n vertices is n-1.

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Similarly, the maximum nonmembership value given to an edge is 1 and the number of edges incident on a vertex can be at most n-1. Hence the maximum nonmembership degree dγ (vi) of any vertex vi in an IFG with n vertices is n-1. 3.9. Proposition: A complete IFG must have at least one pair of vertices whose membership degrees are same and at least one pair of vertices whose nonmembership degrees are same. Proof: Let G = < V, E > be a complete IFG. Case 1: Suppose μ1 (vi) and γ1 (vi) are equal, for all vi ε V, then obviously μ2(vi,vj) are all equal. Hence the degrees of all vertices are equal. Therefore the result is true. Case 2: Suppose μ1(vi) and γ1(vi) are distinct for all vi ε V. We have dμ(vi) = Σ vi ≠ vj μ2(vi,vj) and dγ(vi) = Σ vi≠vj γ2(vi,vj). Since vi are adjacent to vj, we have maximum degree of membership value and minimum degree of nonmembership value of one pair of vertices are equal. 4. Effective Degree 4.1. Definition: An edge e = (x, y) of an IFG G = < V, E > is called an effective edge if μ2(x, y) = μ1(x) Λ μ1(y) and γ2(x, y) = γ1(x) V γ1(y). 4.2. Definition: The effective degree of a vertex ‘v’ is defined by dE (v) = (dEμ (v), dEγ (v)), v V where dEμ (v) is the sum of the membership values of the effective edges incident with v and dEγ (v) is the sum of the nonmembership values of effective edges incident with v. 4.3. Definition: The minimum effective degree of G is δE (G) = (δEμ (G), δEγ (G)) where δEμ (G) = Λ{dEμ(v) / v V} and δEγ (G) = Λ{dEγ(v) / v V} 4.4. Definition: The maximum effective degree of G is ΔE(G) = (ΔEμ(G), ΔEγ(G)) where ΔEμ (G) = V {dEμ (v) / v V} and ΔEγ (G)) = V {dEγ (v) /v V} 4.5. Example: In fig 3, dE (v1) = (0, 0), dE (v2) = (0.2, 0.7) dE (v3) = (0.2, 0.7), dE (v4) = (0,0), δ E (G) = (0, 0), ΔE (G) = (0.2, 0.7). 5. Neighbourhood Degree 5.1. Definition: Let G = < V, E > be an IFG. The neighbourhood of any vertex v is defined as N (v) = (Nμ (v), Nγ (v)) where Nμ(v) = { w V ; μ2 (v,w) = μ1(v) Λ μ1(w)} and Nγ(v) ={w V; γ2(v,w) = γ1(v)Vγ1(w)} and N[v]= N(v)U{v} is called the closed neighbourhood of v. 5.2. Definition: The neighbourhood degree of a vertex is defined as dN (v) = (dNμ (v), dNγ (v)) where dNμ (v) = Σ wε N (v) μ1 (w) and dNγ (v) = Σ wεN (v) γ1(w). 5.3. Definition: The minimum neighbourhood degree is defined as δN(G)=(δNµ(G),δNγ(G)) where δNµ(G)=Λ{dNµ(v): v V} and δNγ(G)= Λ{dN(v):v V} 5.4. Definition: The maximum neighbourhood degree is defined as ΔN(G)=(ΔNµ(G),ΔNγ(G)) where ΔNµ(G)=V{dNµ(v): v V} and ΔNγ(G)= V{dNγ(v):v V} 5.5. Example: In fig 2. dN(v1)=(1.9,0.8), dN (v1)=(2,0.7) dN (v3)=(2.1,0.7) dN (v4)=(1.8,0.8) δN(G)=(1.8,0.7) ΔN(G) = (2.1,0.8) 5.6. Remark: A vertex ‘v’ is an isolated vertex if N(v)=ф 5.7. Definition: The closed neighbourhood degree of a vertex ‘v’ is defined as dN [v]=(dNµ[v],dNγ[v]) where dNµ[v] = Σ wε N (v) μ1 (w) + µ1(v) and dNγ[v] = Σ wεN (v) γ1(w) + γ1(v) 5.8. Definition: The minimum closed neighbourhood degree is defined as δN[G] = (δNµ[G], δNγ[G]) where δNµ[G]= Λ{dNµ[v] : v V} and δNγ[G]= Λ{dNγ[v] :v V} 14

Degree, Order and Size in Intuitionistic Fuzzy Graphs

5.9. Definition: The maximum closed neighbourhood degree is defined ΔN[G]=(ΔNµ[G],ΔNγ[G]) where ΔNµ[G]=V{dNµ[v]:v V} and ΔNγ[G] = V{ΔNγ[v]:v V}

as

6. Regular Fuzzy Graph 6.1. Definition: An Intuitionistic fuzzy graph G = is said to be regular, if all the vertices have the same closed neighbourhood degree. (ie) if δNμ [G] = ΔNμ[G] and δNγ[G] = ΔNγ[G]. 6.2. Example: In fig 2. dN [v1] = dN [v2] = dN [v3] = dN [v4] = (2.6 , 1) δN [G] = ΔN[G] = (2.6, 1). 6.3. Theorem: Every intuitionistic complete fuzzy graph is an intuitionistic regular fuzzy graph. Proof: Let G = < V, E > be a complete IFG then by definition of complete IFG we have μ2 (x,y) = μ1(x) Λ μ2(y) and γ2 (x,y) = γ1(x)V γ1 (y) for every x, y V. By definition, the closed neighbourhood μ degree of each vertex is the sum of the membership values of the vertices and itself and the closed neighbourhood γ degree of each vertex is the sum of the nonmembership values of the vertices and itself. Therefore all the vertices will have the same closed neighbourhood μ degree and closed neighbourhood γ degree. This implies minimum closed neighbourhood degree is equal to maximum closed neighbourhood degree (ie) δNμ [G] = ΔNμ [G] and δNγ [G] =ΔNγ [G]. This implies G is an intuitionistic regular fuzzy graph. Hence the theorem. 7. Order and Size of an IFG 7.1. Definition: Let G = < V, E > be an IFG. Then the order of G is defined to be O(G)=(Oμ(G),Oγ(G) ) where Oμ(G) = Σ v V μ1 (v) and Oγ (G) = Σ v V γ 1 (v) 7.2. Definition: The size of G is defined to be S(G)= (Sμ(G), Sγ(G)) where Sμ(G)= Σu≠v μ2(u,v) and Sγ (G) =Σ u≠v γ2 (u, v). 7.3. Example: In fig 3. O (G) = (2.0, 1.6) S (G) = (1.2, 2.8) 7.4. Proposition: The order of a complete IFG is same as the closed neighbourhood degree of each vertex (i.e) Oμ(G) = ( [v]: v V) and Oγ(G) = ( [v]: v V) Proof: Let G = < V, E > be a complete IFG. The μ-order of G, Oμ(G) is the sum of the membership value of all the vertices and γ-order of G, Oγ(G) is the sum of the nonmembership value of all the vertices. Since G is a complete IFG, the closed neighbourhood μ-degree of each vertex is the sum of the membership value of vertices and the closed neighbourhood γ-degree of each vertex is the sum of the nonmembership value of vertices. Hence the result. 8. Conclusion In this paper, we have analysed some concepts of intuitionistic fuzzy graphs which is an extension of graphs. Much more work could be done to investigate the structure of IFGs. It would be useful since IFGs have applications in network analysis and pattern clustering. References 1. 2.

Atanassov KT. Intuitionistic fuzzy sets: theory and applications. Physica, New York, 1999. Parvathi, R. and Karunambigai, M.G., Intuitionistic Fuzzy Graphs, Computational Intelligence, Theory and applications, International Conference in Germany, Sept 18 -20, 2006.

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3. 4. 5. 6.

Nagoor Gani, A. and Basheer Ahamed, M., Order and Size in Fuzzy Graphs, Bulletin of Pure and Applied Sciences, Vol 22E (No.1) 2003; p.145-148. Bhattacharya, P., Some remarks on fuzzy graphs, Pattern Recognition Letters 6: 297-302, 1987. Harary,F., Graph Theory, Addition Wesley, Third Printing, October 1972. Zimmermann, H.J., Fuzzy Set Theory and its Applications, Kluwer-Nijhoff, Boston, 1985.

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