Ratio Mathematica Volume 46, 2023 Some operations on complex fuzzy graphs Azhagendran.N * Mohamed Ismayil.A † Abstract In this paper we discussed about some types of complex fuzzy graphs which is the extension of fuzzy graph. As the membership value of el- ements of fuzzy graph is in between 0 and 1. In complex fuzzy graph it will be extended to unit circle of complex plane. Therefore we have to consider the amplitude value as well as phase term value. We are considering complex fuzzy graphs in polar forms. Some operations on complex fuzzy graphs such as union, intersection, composition, and Cartesian products are introduced.Some basic theorems with re- spect to the above mentioned operations are proved with examples. 2020 AMS subject classifications:05C62, 05C72, 05C76 1 *Jamal Mohamed College (Autonomous),Bharathidasan University, Trichy, India; e- mail@address.azhagendran87@gmail.com †Jamal Mohamed College (Autonomous),Bharathidasan University, Trichy, India; e- mail@address.amismayil1973@yahoo.co.in 1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20, 2023. DOI: 10.23755/rm.v46i0.1060. ISSN: 1592-7415. eISSN: 2282-8214. ©N.Azhagendran et al. This paper is published under the CC-BY licence agreement. 79 N.Azhagendran and A.Mohamed Ismayil 1 Introduction L.A.Zadeh [1965] gave the idea of fuzzy set which is similar to proba- bility function. The idea of complex fuzzy set (CFS) was first given by Ramot et al. [2002]. Accordingly a CFS is an extension of the fuzzy set whose range is extended from [0,1] to a disc of unit radius in complex plane. Xueling et al. [2019] introduced some basic operations on complex fuzzy set. They have devel- oped a new algorithm in signals and system by using complex fuzzy sets. After that some results have been given by P.Bhattacharya [1987] about fuzzy graphs. Thirunavukarasu et al. [2016] extended the fuzzy graph to complex fuzzy graph. Fuzzy graph were narrated by Mordeson and Peng [1994]. Nagoorgani and Latha [2015] gave some operations such as cartesian product, conjuction, disjunction in fuzzy graph. Shannon and Atanassov [1994] defined intuitionistic fuzzy graphs. After that many authors added their ideas to intuitionistic fuzzy graph. Naveed et al. [2019] introduced the complex intuitionistic fuzzy graphs with certain no- tions of union, join and composition. For Crisp graph we can refer Graph theory by Harary [1969]. In this paper, we introduced some types of complex fuzzy graphs with exam- ples. Also some operations on complex fuzzy graphs such as union, intersection, composition and cartesian product with suitable examples. 2 Premilinaries Definition 2.1. A complex fuzzy graph Gc = (σc,µc) is defined on a graph G = (V,E) is a pair of complex functions σc : V → r(z)eiθ(z),µc : E ⊆ V × V → R(e)eiϕ(e) such that µc(z1,z2) = R(e)eiϕ(e), where R(e) ≤ min{r(z1),r(z2)} and ϕ(e) ≤ min{θ(z1),θ(z2)} for all z1,z2 ∈ V and 0 ≤ r(z1),r(z2) ≤ 1,0 ≤ θ(z1),θ(z2) ≤ 2π. Example 2.1. Consider the complex fuzzy graph Gc = (σc,µc) where σc = {z1/0.2eiπ,z2/0.5ei0.5π,z3/0.7}, µc = {(z1,z2)/0.1ei0.5π,(z1,z3)/0.1,(z2,z3)/0.5} Definition 2.2. A complex fuzzy graph Gc = (σc,µc) is said to be a complement of CFG Gc if i) σc(z) = σc(z) and ii) µc(z1,z2) = R(e)eiϕ(e), where R(e) = min {r(z1),r(z2)} − R(e) and ϕ(e) = min {θ(z1),θ(z2)} − ϕ(e),∀z1,z2 ∈ V . 80 Some operations on complex fuzzy graphs z2(0.5e i0.5π) z3(0.7) e3(0.5) z1(0.2e iπ) e2(0.1) z1(0.2e iπ) e1(0.1e i0.5π) Figure 1: Complex fuzzy graph Example 2.2. Consider the example 2.1, The complement of a CFG given in the figure 1 is given by σc = {z1/0.2eiπ,z2/0.5ei0.5π,z3/0.7} , µc = {(z1,z2)/0.1,(z1,z3)/0.1} Definition 2.3. The order p and size q of a CFG Gc = (σc,µc) defined on G = (V,E) are defined by p = ∑ z∈V r(z).e i ∑ z∈V θ(z) ; q = ∑ e=(zi,zj)∈E R(e).e i ∑ e=(zi,zj)∈E ϕ(e) , where R(e) ≤ min{r(zi),r(zj)} and ϕ(e) ≤ min{θ(zi),θ(zj)} for all zi,zj ∈ V . Example 2.3. Consider the example 2.1, the order of Gc is p = 1.4ei1.5π, the size of Gc is q = 0.7ei0.5π Definition 2.4. The degree of a vertex zi in a CFG Gc = (σc,µc) defined on G = (V,E) is defined by d(zi) = ∑ e=(zi,zj)∈µc R(e) . e i ∑ e=(zi,zj)∈µc ϕ(e) such that µc(zi,zj) = R(e) . eiϕ(e), for all zj ∈ σc. Example 2.4. Consider the example 2.1, d(z1) = 0.2ei0.5π;d(z2) = 0.6ei0.5π; d(z3) = 0.6 Definition 2.5. A CFG is said to be complete for every pair of vertices µc(z1,z2) = R(e)eiϕ(e) where R(e) = min {r(z1),r(z2)} and ϕ(e) = min {θ(z1),θ(z2)} for all z1,z2 ∈ V Example 2.5. Let Gc = (σc,µc) be a CFG,where σc = {z1/0.5ei0.7π,z2/0.8eiπ,z3/0.6eiπ} µc = {(z1,z2)/0.5ei0.7π,(z1,z3)/0.5ei0.7π,(z2,z3)/0.6eiπ} Definition 2.6. A CFG is regular if d(zi) = d(zj) for all zi,zj ∈ σc. Definition 2.7. In a CFG Gc = (σc,µc) for all zi,zj ∈ σc, the neighbourhood of zi is defined by N(zi) = {zj ∈ σc/(zi,zj) ∈ µc} 81 N.Azhagendran and A.Mohamed Ismayil Definition 2.8. A path P in a CFG Gc = (σc,µc) is a sequence of distinct vertices z0,z1,z2, · · ·zn ∈ V (except possibly z0andzn) such that µc(zi−1,zi)= R(ei)e iϕ(ei),R(ei) > 0,ϕ(ei) ≥ 0, i = 1,2, ...,n. Here n is called the length of the path. The consecutive pairs are called edges of the path. The strength of the path in a CFG is defined by µc(zi−1,zi) = min {R(ei)}eiminϕ(ei), i = 1,2,3, ...,n. It is denoted by S(p). Definition 2.9. The strength of connectedness between two vertices zi abd zj which is defined as the maximum amplitude and maximum phase term values of the strength of all paths between zi and zj. In symbol we denote it as µ∞c (zi,zj) = CONNGc(zi,zj), µ ∞ c (zi,zj) = T(e)e iψ(e),0 ≤ T(e) ≤ 1,0 ≤ ψ(e) ≤ 2π, where T(e) is maximum amplitude value of all paths between zi and zj and ψ(e) is max- imum phase term value of all paths between zi and zj. For any arc (zi,zj), if R(e) ≥ T(e) and ϕ(e) ≥ ψ(e) then the arc (zi,zj) is said to be strong. Definition 2.10. The strong degree of a vertex z in a CFG is defined by sum of membership values of strongarcs incident at z ,and it is denoted by ds(z). Definition 2.11. The strong neighbourhood of zi in a CFG is defined by Ns(zi)={zj ∈ V/(zi,zj)is a strong arc}. Definition 2.12. A CFG Gc = (σc,µc) is said to be bipartite if the vertex σc can be partitioned into two non-empty sets σc1 and σc2 such that µc(zi,zj) = 0 if zi,zj ∈ σc1 and zi,zj ∈ σc2. Definition 2.13. A CFG Gc = (σc,µc) is said to be complete bipartite if the vertex σc can be partitioned into two non-empty sets σc1 and σc2 such that µc(zi,zj) = R(e).eiϕ(e), where R(e) = min {r(zi),r(zj)} ,ϕ(e) = min {θ(zi),θ(zj)} for zi ∈ σc1 and zj ∈ σc2. Definition 2.14. A vertex zi of a complex fuzzy graph Gc = (σc,µc) is said to be an isolated vertex if µc(zi,zj) = 0,∀zj ∈ V − {zi} ,(i.e)N(zi) = ∅. 3 Operations on complex fuzzy graph In this section the operations union, intersection, composition and cartesian product of CFGs are defined with examples. Some propositions based on the above operations are stated and proved. Definition 3.1. Let the two complex fuzzy graphs Gc1 = (σc1,µc1), and Gc2 = (σc2,µc2) defined on two graphs G1 = (V1,E1) and G2 = (V2,E2) respectively. Let Gc1 be a pair of complex functions σc1 : V1 → r1(z)eiθ1(z),µc1 : E1 ⊆ 82 Some operations on complex fuzzy graphs V1 × V1 → R1(e).eiϕ1(e) such that µc(z1,z2) = R1(e)eiϕ1(e) where R1(e) ≤ min {r1(z1),r1(z2)} ,ϕ1(e) ≤ min {θ1(z1),θ1(z2)}. Also Gc2 is a pair of com- plex functions σc2 : V2 → r2(z)eiθ2(z),µc2 : E2 ⊆ V2 × V2 → R2(e).eiϕ2(e) such that µc(z1,z2) = R2(e)eiϕ2(e) where R2(e) ≤ min {r2(z1),r2(z2)} ,ϕ2(e) ≤ min {θ2(z1),θ2(z2)}. Then the union of two complex fuzzy graphs Gc = (σc = σc1 ∪ σc2,µc = µc1 ∪ µc2) on G = (V = V1 ∪ V2,E = E1 ∪ E2) is defined as follows (i) σc(z) = (σc1∪σc2)(z) =   r1(z)e iθ1(z),∀z ∈ V1and z /∈ V2 r2(z)e iθ2(z),∀z ∈ V2and z /∈ V1 max {r1(z),r2(z)}eimax{θ1(z),θ2(z)},z ∈ V1 ∩ V2 (ii) µc(z1,z2) = (µc1 ∪ µc2)(z1,z2) =  R1(e)e iϕ1(e),∀ (z1,z2) ∈ E1 and(z1,z2) /∈ E2 R2(e)e iϕ2(e),∀ (z1,z2) ∈ E2 and (z1,z2) /∈ E1 max {R1(e),R2(e)}eimax{ϕ1(e),ϕ2(e)},(z1,z2) ∈ E1 ∩ E2 Proposition 3.1. Prove that the union of two complex fuzzy graph is also a com- plex fuzzy graph. Proof. Let Gc1 and Gc2 be two CFGs, then the union Gc = Gc1 ∪Gc2 is discussed in three different case. For vertices, i) Suppose that z ∈ V1 and z /∈ V2 then (σc1 ∪ σc2)(z) = max {r1(z),r2(z)} .eimax{θ1(z),θ2(z)} = max {r1(z),0} .eimax{θ1(z),0} = r1(z).e iθ1(z),∀z ∈ V1and z /∈ V2. ii) Similarly we can prove, for z /∈ V1 and z ∈ V2 iii) Suppose that z ∈ V1 ∩ V2, then (σc1 ∪ σc2)(z) = max {σc1,σc2} = max { r1(z)e iθ1(z).r2(z)e iθ2(z) } = max {r1(z),r2(z)} .eimax{θ1(z),θ2(z)},∀z ∈ V1 ∩ V2. For edges, i) Suppose that (z1,z2) ∈ E1,(z1,z2) /∈ E2 (µc1 ∪ µc2)(z1,z2) = max {µc1(z1,z2),µc2(z1,z2)} ≤ max {min {r1(z1),r1(z2),min {r2(z1),r2(z2)}}} . eimax{min{θ1(z1),θ1(z2)},min{θ2(z1),θ2(z2)}} = max {R1(e),R2(e)} .eimax{ϕ1(e),ϕ2(e)} = max {R1(e),0} .eimax{ϕ1(e),0} = R1(e).e iϕ1(e),∀(z1,z2) ∈ E1,(z1,z2) /∈ E2 83 N.Azhagendran and A.Mohamed Ismayil ii) Similarly, we can prove that, (z1,z2) /∈ E1,(z1,z2) ∈ E2 iii) Suppose that (z1,z2) ∈ E1 ∩ E2 (µc1 ∪ µc2)(z1,z2) = max {µc1(z1,z2),µc2(z1,z2)} ≤ max {min {r1(z1),r1(z2)} ,min {r2(z1),r2(z2)}} . eimax{min{θ1(z1),θ1(z2)},min{θ2(z1),θ2(z2)}} = max {R1(e),R2(e)} .eimax{ϕ1(e),ϕe(z)} for (z1,z2) ∈ E1 ∩ E2 Therefore the union of two CFGs is also a CFG. Example 3.1. Consider the two complex fuzzy graphs Gc1 = (σc1,µc1) and Gc2 = (σc2,µc2) given below σc1 = {z1/0.4ei0.8π,z2/0.7ei2π,z3/0.5ei1.2π} , µc1 = {(z1,z2)/0.4ei0.8π,(z2,z3)/0.4eiπ} and σc2 = {z1/0.5eiπ,z3/0.5ei1.4π, z4/0.8e i2π, µc2 = {(z1,z4)/0.4ei0.6π,(z3,z4)/0.5ei1.4π} Then the union Gc = (σc1 ∪ σc2 = σc,µc1 ∪ µc2 = µc) on G = (V1 ∪ V2,E1 ∪ E2) can be written as σc = {z1/0.5eiπ,z2/0.7ei2π,z3/0.5ei1.4π,z4/0.8ei2π} µc = {(z1,z2)/0.4ei0.8π,(z1,z4)/0.4ei0.6π,(z3,z4)/0.5ei1.4π,(z2,z4)/0.4eiπ} z1(0.4e i0.8π) z2(0.7e i2π) 0.4ei0.8π z3(0.5e i1.2π) 0.4eiπ Figure 2: Gc1 z1(0.5e iπ) z4(0.8e i2π) 0.4ei0.6π z3(0.5e i1.4π) 0.5ei1.4π Figure 3: Gc2 z2(0.7e i2π) z1(0.5e iπ) 0.4ei0.8π z3(0.5e i1.4π) 0.4eiπ z4(0.8e i2π) z1(0.5e iπ) z4(0.8e i2π) z3(0.5e i1.4π) 0.4ei0.6π 0.5ei1.4π Figure 4: Gc1 ∪ Gc2 Remark 3.1. Union of two strong CFG need not be a strong CFG. 84 Some operations on complex fuzzy graphs Example 3.2. Consider the Strong CFGs Gc1 = (σc1,µc1) and Gc2 = (σc2,µc2) where σc1 = {z1/0.4ei2π,z2/0.6eiπ}, µc1 = {(z1,z2)/0.4eiπ} and σc2 = {z1/0.8ei0.2π,z2/0.2eiπ}, µc2 = {(z1,z2)/0.2ei0.2π} then the union is given by σc = {z1/0.8ei2π,z2/0.6eiπ}, µc = {(z1,z2)/0.4ei0.2π}. However this is not a strong CFG. Definition 3.2. Let the two complex fuzzy graphs Gc1 = (σc1,µc1), and Gc2 = (σc2,µc2) defined on two graphs G1 = (V1,E1) and G2 = (V2,E2) respectively. Let Gc1 be a pair of complex functions σc1 : V1 → r1(z)eiθ1(z),µc1 : E1 ⊆ V1 × V1 → R1(e).eiϕ1(e) such that µc(z1,z2) = R1(e)eiϕ1(e) where R1(e) ≤ min {r1(z1),r1(z2)} ,ϕ1(e) ≤ min {θ1(z1),θ1(z2)}. Also Gc2 is a pair of com- plex functions σc2 : V2 → r2(z)eiθ2(z),µc2 : E2 ⊆ V2 × V2 → R2(e).eiϕ2(e) such that µc(z1,z2) = R2(e)eiϕ2(e) where R2(e) ≤ min {r2(z1),r2(z2)} ,ϕ2(e) ≤ min {θ2(z1),θ2(z2)}. Then the intersection of two complex fuzzy graphs Gc = (σc = σc1 ∩ σc2,µc = µc1 ∩ µc2) is defined as follows (i) σc(z) = (σc1 ∩ σc2)(z) = min {r1(z),r2(z)}eimin{θ1(z),θ2(z)},z ∈ V1 ∩ V2 (ii) µc(z1,z2) = (µc1 ∩ µc2)(z1,z2) ={ min {R1(e),R2(e)}eimin{ϕ1(e),ϕ2(e)},(z1,z2) ∈ E1 ∩ E2 0,otherwise Proposition 3.2. Prove that the intersection of two complex fuzzy graphs is also a complex fuzzy graph. Proof. Let Gc1 and Gc2 be two complex fuzzy graphs, then i) (σc1 ∩ σc2)(z) = min {r1(z),r2(z)} .eimin{θ1(z),θ2(z)} Itistrivial. ii) For (z1,z2) ∈ E1 ∩ E2 (µc1 ∩ µc2)(z1,z2) = min {µc1(z1,z2),µc2(z1,z2)} .eimin{ϕ1(z),ϕ2(z)} ≤ min {min {r1(z1),r1(z2)} ,min {r2(z1),r2(z2)}} . eimin{min{θ1(z1),θ1(z2)},min{θ2(z1),θ2(z2)}} = min {min {r1(z1),r2(z1)} ,min {r1(z2),r2(z2)}} . eimin{min{θ1(z1),θ1(z2)},min{θ2(z1),θ2(z2)}} = min {R1(z),R2(z)} .eimin{ϕ1(z),ϕ2(z)} where R1(z) ≤ min {r1(z1),r2(z1)} ,R2(z) ≤ min {r1(z2),r2(z2)} , ϕ(z1) ≤ min {θ1(z1),θ2(z1)} , ϕ(z2) ≤ min {θ2(z1),θ2(z2)} Hence the Gc1 ∩ Gc2 is a CFG. 85 N.Azhagendran and A.Mohamed Ismayil Example 3.3. Consider the two complex fuzzy graphs Gc1 = (σc1,µc1) where σc1 = {z1/0.5ei0.7π,z2/0.8,z3/0.7ei1.5π} ;µc1 = {(z1,z2)/0.5,(z2,z3)/0.7} and Gc2 = (σc2,µc2) where σc2 = {z1/0.4ei2π,z2/0.6ei0.8π,z4/0.7eiπ} ; µc2 = {(z1,z4)/0.4eiπ,(z1,z2)/0.4ei0.8π} Then the intersection of Gc1 and Gc2 on a pair G = (V1 ∩V2,E1 ∩E2) is given by V = {z1/0.4ei0.7π,z2/0.6} ,E = {(z1,z2)/0.4} where V = V1∩V2;E = E1∩E2. Remark 3.2. Intersection of two strong CFGs is also a strong CFG. Definition 3.3. Let the two complex fuzzy graphs Gc1 = (σc1,µc1), and Gc2 = (σc2,µc2) defined on two graphs G1 = (V1,E1) and G2 = (V2,E2) respectively. Let Gc1 be a pair of complex functions σc1 : V1 → r1(a)eiθ1(a),µc1 : E1 ⊆ V1 × V1 → R1(e).eiϕ1(e) such that µc(a1,a2) = R1(e)eiϕ1(e) where R1(e) ≤ min {r1(a1),r1(a2)} ,ϕ1(e) ≤ min {θ1(a1),θ1(a2)}. Also Gc2 is a pair of com- plex functions σc2 : V2 → r2(a)eiθ2(a),µc2 : E2 ⊆ V2 × V2 → R2(e).eiϕ2(e) such that µc(a1,a2) = R2(e)eiϕ2(e) where R2(e) ≤ min {r2(a1),r2(a2)} ,ϕ2(e) ≤ min {θ2(a1),θ2(a2)}. Then the composition of two complex fuzzy graphs is de- fined as follows (i) (σc1 ◦ σc2)(a1,a2) = min {r1(a1),r2(a2)}eimin{θ1(a1),θ2(a2)},∀a1,a2 ∈ V = V1 ◦ V2 (ii) (µc1 ◦ µc2)((a,a2),(a,b2)) = min {r1(a),R2(a)}eimin{θ1(a),ϕ2(a)} where R2(a) ≤ min {r2(a2),r2(b2)} ;ϕ2(a) ≤ min {θ2(a2),θ2(b2)} ,∀a ∈ V1,(a2,b2) ∈ E2 (iii) (µc1 ◦ µc2)((a1,a),(b1,a)) = min {R1(a),r2(a)}eimin{ϕ1(a),θ2(a)} whereR1(a) ≤ min {r1(a1),r1(b1)} ;ϕ1(a) ≤ min {θ1(a1),θ1(b1)} ,∀a ∈ V2,(a1,b1) ∈ E1 (iv) (µc1 ◦ µc2)((a1,a2),(b1,b2)) = min {r2(a2),r2(b2),R1(a)} . eimin{θ2(a2),θ2(b2),ϕ1(a)} ,∀a2,b2 ∈ V2,a2 ̸= b2,(a1,b1) ∈ E1, whereR1(a) ≤ min {r1(a1),r1(b1)} ;ϕ2(a) ≤ min {θ1(a1),θ1(b1)} , Proposition 3.3. Prove that the composition of two complex fuzzy graphs is also a complex fuzzy graph. Proof. Let Gc1 and Gc2 be two CFGs then we prove that GC1 ◦ GC2 is a CFG. (i) (σc1 ◦ σc2) = (a1,a2) = min {r1(a1),r2(a2)}eimin{θ1(a1),θ2(a2)},∀a1,a2 ∈ V . It is trivial. 86 Some operations on complex fuzzy graphs (ii) (µc1 ◦ µc2)((a,a2)(a,b2)) = min {r1(a),R2(a)}eimin{θ1(a),θ2(a)} ≤ min {r1(a),min {r2(a2),r2(b2)}}eimin{θ1(a),min{θ2(a2),θ2(b2)}} = min {min {r1(a),r2(a2)} ,min {r1(a),r2(b2)}} . eimin{min{θ1(a),θ2(a2)},min{θ1(a),θ2(b2)}} = min {r(a1),r(b1)}eimin{θ(a1),θ(b1)} , where r(a1) = min {r1(a),r2(a2)} , r(b1) = min {r1(a),r2(b2)} ,θ(a1) = min {θ1(a),θ2(a2)} , θ(b1) = min {θ1(a),θ2(b2)} for all a ∈ V1and(a2,b2) ∈ E2. (iii) (µc1 ◦ µc2)((a1,b)(b1,b)) = min {R1(a),r2(b)}eimin{θ1(a),θ2(b)} ≤ min {min {r1(a1),r1(b1)} ,r2(b)}eimin{min{θ1(a1),θ1(b1)},θ2(b)} = min {min {r1(a1),r2(b)} ,min {r1(b1),r2(b)}}. eimin{min{θ1(a1),θ2(b)},min{θ1(b1),θ2(b)}} = min {r(a2),r(b2)}eimin{θ(a2),θ(b2)}, where r(a2) = min {r1(a1),r2(b)} , r(b2) = min {r1(b1),r2(b)} ,θ(a2) = min {θ1(a1),θ2(b)} , θ(b2) = min {θ1(b1),θ2(b)} for all b ∈ V2 and (a1,b1) ∈ E1. (iv) For all a2,b2 ∈ V2,a2 ̸= b2,(a1,b1) ∈ E1 (µc1 ◦ µc2)((a1,a2),(b1,b2)) = min {r2(a2),r2(b2),R1(a)} . eimin{θ2(a2),θ2(b2),ϕ(a)} ≤ min {r2(a2),r2(b2),min {r1(a1),r1(b1)}} . eimin{θ2(a2),θ2(b2),min{θ1(a1),θ1(b1)}} = min {r2(a2),r2(b2),r1(a1),r1(b1)} .eimin{θ2(a2),θ2(b2),θ1(a1),θ1(b1)} = min {r(a),r(b)} .eimin{θ(a),θ(b)}. where r(a) = min {r1(a1),r2(a2)} ,r(b) = min {r1(b1),r2(b2)} θ(a) = min {θ1(a1),θ2(a2)} ,θ(b) = min {θ1(b1),θ2(b2)}. Hence GC1 ◦ GC2 is a CFG. Example 3.4. Consider the CFGs, Gc1 = (σc1,µc1) where σc1 = {z1/0.4ei0.7π,z2/0.6ei2π} ;µc1 = {(z1,z2)/0.4ei0.5π} and Gc2 = (σc2,µc2) where σc2 = {z3/0.5ei1.2π,z4/0.6ei0.7π} ;µc2 = {(z3,z4)/0.5ei0.7π} . Then the composition of Gc1 and Gc2 is given by Gc = (σc,µc) where σc = {z1z3/0.4ei0.7π,z1z4/0.4ei0.7π,z2z3/0.5ei1.2π,z2z4/0.6ei0.7π} , µc = {(z1z3,z1z4)/0.4ei0.7π,(z1z4,z2z4)/0.4ei0.5π,(z2z3,z2z4)/0.5ei0.7π, (z1z3,z2z3)/0.4e i0.5π,(z1z3,z2z4)/0.4e i0.5π,(z1z4,z2z3)/0.4e i0.5π } Remark 3.3. The Composition of two strong CFG is need not be a strong CFG. Definition 3.4. Let the two complex fuzzy graphs Gc1 = (σc1,µc1), and Gc2 = (σc2,µc2) defined on two graphs G1 = (V1,E1) and G2 = (V2,E2) respectively. Let Gc1 be a pair of complex functions σc1 : V1 → r1(a)eiθ1(a),µc1 : E1 ⊆ V1 × V1 → R1(e).eiϕ1(e) such that µc(a1,a2) = R1(e)eiϕ1(e) where R1(e) ≤ 87 N.Azhagendran and A.Mohamed Ismayil min {r1(a1),r1(a2)} ,ϕ1(e) ≤ min {θ1(a1),θ1(a2)}. Also Gc2 is a pair of com- plex functions σc2 : V2 → r2(a)eiθ2(a),µc2 : E2 ⊆ V2 × V2 → R2(e).eiϕ2(e) such that µc(a1,a2) = R2(e)eiϕ2(e) where R2(e) ≤ min {r2(a1),r2(a2)} ,ϕ2(e) ≤ min {θ2(a1),θ2(a2)}. Then the Cartesian product of two complex fuzzy graphs is defined as follows (i) (σc1 × σc2)(a1,a2) = min {r1(a1),r2(a2)}emin{θ1(a1),θ2(a2)},fora1,a2 ∈ V (ii) (µc1 × µc2)((a,a2),(a,b2)) = min {r1(a),R2(a)} .emin{θ1(a),ϕ2(a)} where R2(a) ≤ min {r2(a2),r2(b2)} ;ϕ2(a) ≤ {θ2(a2),θ2(b2)} for all a ∈ V1 and (a2,b2) ∈ E2 (iii) (µc1 × µc2)((a1,a),(b1,a)) = min {R1(a),r2(a)} .emin{ϕ1(a),θ2(a)} where R1(a) ≤ min {r1(a1),r1(b1)} ;ϕ2(a) ≤ {θ1(a1),θ1(b1)} for all a ∈ V2 and (a1,b1) ∈ E1 Proposition 3.4. The cartesian product of two complex fuzzy graphs is also a complex fuzzy graph. Proof. Let Gc1 and Gc2 be two complex fuzzy graphs, then we prove that Gc1 × Gc2 is a CFG. (i) σc1 × σc2(a1,a2) = min {r1(a1),r2(a2)} .eimin{θ1(a1,θ2(a2))},∀a1,a2 ∈ V1 × V2. It is trivial, we have to verify the conditions only for E1 × E2 (ii) (µc1 × µc2)((a,a2),(a,b2)) = min {r1(a),R2(a)} .eimin{θ1(a),ϕ2(a)} ≤ min {r1(a),min {r2(a2),r2(b2)}} .eimin{θ1(a),min{θ2(a2),θ2(b2)}} = min {min {r1(a),r2(a2)} ,min {r1(a),r2(b2)}} . eimin{min{θ1(a),θ2(a2)},min{θ1(a),θ2(b2)}} = min {r(a1),r(b1)} .eimin{θ(a1),θ(b1)} where r(a1) = min {r1(a),r2(a2)} ,r(b1) = min {r1(a),r2(b2)} , θ(a1) = min {θ1(a),θ2(a2)} ,θ(b1) = min {θ1(a),θ2(b2)} forall a ∈ V1,(a2,b2) ∈ E2 (iii) (µc1 × µc2)((a1,a),(b1,a)) = min {R1(a),r2(a)} .eimin{ϕ1(a),θ2(a)} ≤ min {min {r1(a1),r1(b1)} ,r2(a)} .eimin{min{θ1(a1),θ1(b1)},θ2(a)} = min {min {r1(a1),r2(a)} ,min {r1(b1),r2)}} . eimin{min{θ1(a1),θ2(a)},min{θ1(b1),θ2(a)}} = min {r(a2),r(b2)} .eimin{θ(a2),θ(b2)} where r(a2) = min {r1(a1),r2(a)} ,r(b2) = min {r1(b1),r2(a)} , θ(a2) = min {θ1(a1),θ2(a)} ,θ(b2) = min {θ1(b1),θ2(a)} forall a ∈ V2,(a1,b1) ∈ E1 Hence Gc1 × Gc2 is a CFG. 88 Some operations on complex fuzzy graphs Example 3.5. Let Gc1 and Gc2 be two complex fuzzy graphs. σc1 = {z1/0.4ei2π,z2/0.5ei1.2π} , µc1 = {(z1,z2)/0.4ei1.2π} , σc2 = {z3/0.5ei0.7π,z4/0.8eiπ, z5/0.4ei1.7π, µc2 = {(z3,z4)/0.4ei0.7π,(z4,z5)/0.4eiπ}. Then the cartesian product is given by σc1 × σc2 = {z1z3/0.4ei0.7π,z1z4/0.4eiπ,z1z5/0.4ei1.7π, z2z3/0.5e i0.7π,z2z4/0.5e iπ,z2z5/0.4e i1.2π } µc1 × µc2 = {(z1z3,z1z4)/0.4ei0.7π, (z1z4,z1z5)/0.4e iπ,(z1z3,z2z3)/0.4e i0.7π,(z2z3,z2z4)/0.5e i0.7π, (z2z4,z2z5)/0.4e i1.2π,(z1z4,z2z4)/0.4e iπ,(z1z5,z2z5)/0.4e i1.2π } z1(0.4e i1.2π) z2(0.5e i1.2π) 0.4ei1.2π Figure 5: Gc1 z3(0.5e i0.7π) z4(0.8e iπ) 0.4ei0.7π z5(0.4e i1.7π) 0.4eiπ Figure 6: Gc2 (z1,z3)(0.4e i0.7π) (z1,z4)(0.4e iπ) 0.4ei0.7π (z1,z5)(0.4e i1.7π) 0.4eiπ (z2,z3)(0.5e i0.7π) (z2,z4)(0.5e iπ) 0.4ei0.7π (z2,z5)(0.4e i1.2π) 0.4eiπ 0.4ei0.7π 0.4eiπ 0.4ei1.2π Figure 7: Gc1 × Gc2 Remark 3.4. Cartesian product of two strong CFG is also a strong CFG. 4 Conclusions In this paper, we discussed about some operations on complex fuzzy graphs such as union, intersection, composition and Cartesian product with examples. 89 N.Azhagendran and A.Mohamed Ismayil Complex fuzzy graph is an extension of a fuzzy graph. We are working to extend the algorithms applied to complex fuzzy graphs are i) Domination on complex fuzzy graphs ii) Connectivity of complex fuzzy graphs and more operations on Complex fuzzy graphs References Harary. Graph Theory. Addition wesley, London, 1969. L.A.Zadeh. Fuzzy sets. Information and control, 8(3):338–353, 1965. J. Mordeson and C. Peng. Operations on fuzzy graphs. Information Sci., 79: 159–170, 1994. A. Nagoorgani and R. Latha. Some properties on operations of fuzzy graphs. Advances in Fuzzy sets and Systems, 19(1):1–24, 2015. Y. Naveed, G. Muhammad, K. Seifedine, and A. W. Hafiz. 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