Table Of Content

Bull. Comput. Appl. Math.,Vol.6,No.2,2018 ISSN2244-8659 www. ompama. o.usb.ve Complex neutrosophi graphs 12 3 Naveed Yaqoob , Muhammad Akram CompAMaVol.6,No.2,pp.85-109,2018-A epted De ember22,2018 Abstra t In this resear h arti le, we introdu e the notion of omplex neutrosophi graphs cn cn ( -graphs, for short) and dis uss some basi operations related to -graphs. We des ribe these operations with some examples. We also present energy of omplex neutrosophi graphs. Keywords: Complex neutrosophi sets, omplex neutrosophi graphs, energy. 1 Introdu tion Fuzzy set theory was introdu ed by Zadeh [1℄. Appli ations of these sets have been broadly studied in other aspe ts su h as ontrol [2℄, reasoning [1℄, pattern re ognition [2℄, engineering [3℄, et . Atanassov [4℄ proposed the extended form of fuzzy set by adding a new omponent, alled intuitionisti fuzzy sets (IFSs). The idea of IFSs is more meaningful as well as intensive due to the presen e of degree of truth and falsity membership. Smaranda he [5,6℄ introdu ed the thought of neutrosophi sets by ombining the non- standard analysis. Neutrosophi set theory is applied to image segmentation [7℄, topology [8℄, de ision making [9℄, roboti s [10℄, physi s [11℄ and in many more real life problems. See also [12(cid:21)16℄. 1 Department of Mathemati s, College of S ien e Al-Zul(cid:28), Majmaah University, Al- Zul2(cid:28), Saudi Arabia (na.yaqoobmu.edu.sa, nayaqoobymail. om). 3Correspondingauthor. Department of Mathemati s, University of the Punjab, New Campus, Lahore, Pakis- tan (m.akrampu it.edu.pk). 85 Complex neutrosophi graphs 86 Bu kley [17℄ and Nguyen et al. [18℄ ombined omplex numbers with fuzzy sets. On the other hand, Ramot et al. [19,20℄ extended the range of membership to (cid:16)unit ir le in the omplex plane(cid:17), unlike the others who [0,1] limitedthe range to . Further this on ept has been studied in IFSs [21℄ and Samaranda he's neutrosophi sets [22℄. Agraph isamathemati alobje t ontainingpoints(verti es) and onne - tions (edges), and is a onvenient way of interpreting information involving the relationship between di(cid:27)erent obje ts. However, due to some reasons, in pra ti al appli ations of graph theory, di(cid:27)erent types of un ertainties are frequently en ountered. To handle these un ertainties, Kaufmann [23℄ introdu ed the theory of fuzzy graphs based on Zadeh's fuzzy relations. Later, Rosenfeld [24℄ put forward another elaborated de(cid:28)nition of fuzzy graph with fuzzy vertex and fuzzy edges and developed the stru ture of fuzzy graphs. Mordeson and Peng [25℄ de(cid:28)ned some operations on fuzzy graphs. All the on epts on risp graph theory do not have similarities in fuzzy graphs. Dhavaseelan et al. [26℄ de(cid:28)ned strong neutrosophi graphs. Akram and Shahzadi[27℄(cid:28)rst studied single-valuedneutrosophi graphs. Furtherseveral new on epts on neutrosophi graphs with their appli ations were dis ussed in [28(cid:21)31℄. On the other hand, Akram and Shahzadi [29℄ have shown that there are some (cid:29)aws in Broumi et al. [30℄'s de(cid:28)nitions. His de(cid:28)nitions are not useful for the study of applied network models. Re ently, Akram and Naz [32℄ determined the energy of Pythagorean fuzzy graphs. In this pa- per, we provide the new on ept of omplex neutrosophi graphs with some fundamental operations. We also des ribe energy of omplex neutrosophi graphs. 2 Preliminaries and basi de(cid:28)nitions U = ∅. De(cid:28)nition 1. [19℄ Let 6 A omplex fuzzy set (CFS) A, is an obje t of the form = (x,µ (x)) : x U = (x,u (x)eiωA(x)) : x U , A A A { ∈ } { ∈ } i = √ 1, u (x) [0,1] 0 < ω (x) < 2π. A A where − ∈ and U = ∅. cn De(cid:28)nition 2. [22℄ Let 6 A omplex neutrosophi set ( -set) A, is an obje t of the form = (x,T (x),I (x),F (x)) : x U A A A A { ∈ } = (x,s (x)eiαA(x),t (x)eiβA(x),u (x)eiγA(x)) : x U , A A A { ∈ } 87 Yaqoob, Akram i = √ 1, s (x),t (x),u (x) [0,1] α (x),β (x),γ (x) [0,2π] A A A A A A where − ∈ , ∈ 0− s (x)+t (x)+u (x) 3+. A A A and ≤ ≤ cn , De(cid:28)nition 3. [22℄ Let A and B be two -sets in X where = (x,T (x),I (x),F (x)) : x A A A A { ∈ X} = (x,T (x),I (x),F (x)) : x . B B B and B { ∈ X} x , Then for all ∈ X s (x) < s (x), t (x) > t (x), u (x) > u (x) A B A B A B (1) A ⊂ B if and only if for α (x) < α (x), β (x) > β (x), γ (x) > γ (x) A B A B A B amplitude terms and for phase terms. = s (x) = s (x), t (x) = t (x), u (x) = u (x) A B A B A B (2) A B if and only if for α (x) = α (x), β (x) = β (x), γ (x) = γ (x) A B A B A B amplitude terms and for phase terms. = (x,T (x),I (x),F (x)) : x A∪B A∪B A∪B (3) A∪B { ∈ X} where T (x) = s (x)eiαA∪B(x) = max s (x),s (x) eimax{αA(x),αB(x)}, A∪B A∪B A B { } I (x) = t (x)eiβA∪B(x) = min t (x),t (x) eimin{βA(x),βB(x)}, A∪B A∪B A B { } F (x) = u (x)eiγA∪B(x) = min u (x)u (x) eimin{γA(x),γB(x)}. A∪B A∪B A B { } = (x,T (x),I (x),F (x)) : x A∩B A∩B A∩B (4) A∩B { ∈ X} where T (x) = s (x)eiαA∩B(x) = min s (x),s (x) eimin{αA(x),αB(x)}, A∩B A∩B A B { } I (x) = t (x)eiβA∩B(x) = max t (x),t (x) eimax{βA(x),βB(x)}, A∩B A∩B A B { } F (x) = u (x)eiγA∩B(x) = max u (x)u (x) eimax{γA(x),γB(x)}. A∩B A∩B A B { } cn De(cid:28)nition 4. Let A and B be two -sets in X, where = (x,T (x),I (x),F (x)) : x A A A A { ∈ X} and = (x,T (x),I (x),F (x)) : x B B B B { ∈ X}. T (x) T (x) I (x) I (x) F (x) F (x) A B A B A B Then ≤ , ≥ , ≥ if and only if T (x) T (x) I (x) I (x) F (x) F (x) A B A B A B | | ≤ | |, | | ≥ | |, | | ≥ | | and α (x) α (x) β (x) β (x) γ (x) γ (x) A B A B A B ≤ , ≥ , ≥ . Complex neutrosophi graphs 88 3 Complex neutrosophi graphs cn = ∅ cn De(cid:28)nition 5. A -relation on X 6 , is a -subset of X ×X of the form = (xy,T (xy),I (xy),F (xy)) : xy B B B B { ∈ X ×X} su h that T (xy) = s (xy)eiαB(xy), I (xy) = t (xy)eiβB(xy), F (xy) = u (xy)eiγB(xy) B B B B B B x,y for all ∈ X. cn = ∅, G = ( , ) De(cid:28)nition 6. A -graph on X 6 is a pair A B where, A is a cn cn -set on X and B is a -relation in X su h that s (xy)eiαB(xy) min s (x),s (y) eimin{αA(x),αA(y)}, B A A ≤ { } t (xy)eiβB(xy) min t (x),t (y) eimin{βA(x),βA(y)}, B A A ≤ { } u (xy)eiγB(xy) max u (x),u (y) eimax{γA(x),γA(y)} B A A ≤ { } x,y for all ∈ X. A and B are alled thGe omplex neutrosophi vertex set and the omplex neutrosophi edge set of , respe tively. Here B is the omplex neutrosophi relation on A. G∗ = ( , ) = a,b,c = Example 1. Consider a graph V E su h that X { }, E ab,bc,ac cn cn { }. Let A be a -subset of X and let B be a -subset of E ⊆ X×X, as given: (0.2ei π/2,0.4ei 2π/5,0.5ei π/4), (0.5ei 3π/4,0.6ei π/2,0.1ei π/8), = a b . A (0.4ei π/2,0.5ei π,0.2ei π/3) ! c (0.1ei π/2,0.4ei π/3,0.3ei π/6), (0.2ei 2π/5,0.4ei 2π/5,0.1ei π/4), = ab bc . B (0.1ei π/4,0.3ei π/4,0.4ei 2π/7) ! ac By routine al ulations, it an be observed that the graph shown in Fig. 1 is cn a -graph. G G cn 1 2 De(cid:28)niGtion 7G. T=he( Cartesia,n produ )t × of two -graphs is de(cid:28)ned as 1 2 1 2 1 2 a pair × A ×A B ×B , su h that: 1. sA1×A2(x1,x2)eiαA1×A2(x1,x2) = min{sA1(x1),sA2(x2))}eimin{αA1(x1),αA2(x2)}, tA1×A2(x1,x2)eiβA1×A2(x1,x2) = min{tA1(x1),tA2(x2))}eimin{βA1(x1),βA2(x2)}, uA1×A2(x1,x2)eiγA1×A2(x1,x2) = max{uA1(x1),uA2(x2))}eimax{γA1(x1),γA2(x2)}, x x 1 2 for all , ∈ X, 89 Yaqoob, Akram cn Fig. 1: -graph 2. sB1×B2((x,x2)(x,y2))eiαB1×B2((x,x2)(x,y2)) = min{sA1(x),sB2(x2y2))}eimin{αA1(x),αB2(x2y2)}, tB1×B2((x,x2)(x,y2))eiβB1×B2((x,x2)(x,y2)) = min{tA1(x),tB2(x2y2))}eimin{βA1(x),βB2(x2y2)}, uB1×B2((x,x2)(x,y2))eiγB1×B2((x,x2)(x,y2)) = max{uA1(x),uB2(x2y2))}eimax{γA1(x),γB2(x2y2)}, x x y 1 2 2 2 for all ∈ X , and ∈ E , 3. sB1×B2((x1,z)(y1,z))eiαB1×B2((x1,z)(y1,z)) = min{sB1(x1y1),sA2(z))}eimin{αB1(x1y1),αA2(z)}, tB1×B2((x1,z)(y1,z))eiβB1×B2((x1,z)(y1,z)) = min{tB1(x1y1),tA2(z))}eimin{βB1(x1y1),βA2(z)}, uB1×B2((x1,z)(y1,z))eiγB1×B2((x1,z)(y1,z)) = max{uB1(x1y1),uA2(z))}eimax{γB1(x1y1),γA2(z)}, z x y 2 1 1 1 for all ∈ X , and ∈ E . cn Example 2. Consider the two G-grapGhs, as shown in Fig. 2. Then, their 1 2 orresponding Cartesian produ t × is shown in Fig. 3. cn cn Proposition 1. The Cartesian produ t of two -graphs is a -graph. Complex neutrosophi graphs 90 cn G G 1 2 Fig. 2: -graphs and cn G G 1 2 Fig. 3: -graph × 91 Yaqoob, Akram 1 2 Proof: The onditions for A × A are obvious, therefore, we verify only 1 2 onditions for B ×B . x x y 1 2 2 2 Let ∈ X , and ∈ E . Then sB1×B2((x,x2)(x,y2))eiαB1×B2((x,x2)(x,y2)) = min{sA1(x),sB2(x2y2))}eimin{αA1(x),αB2(x2y2)} ≤ min{sA1(x),min{sA2(x2),sA2(y2)}}eimin{αA1(x),min{αA2(x2),αA2(y2)}} min α (x),α (x ) , imin { A1 A2 2 }  min s (x),s (x ) ,  min α (x),α (y )  = min { A1 A2 2 } e { A1 A2 2 } min s (x),s (y )   (cid:26) { A1 A2 2 } (cid:27) = min{sA1×A2(x,x2),sA1×A2(x,y2)}eimin{αA1×A2(x,x2),αA1×A2(x,y2)}, tB1×B2((x,x2)(x,y2))eiβB1×B2((x,x2)(x,y2)) = min{tA1(x),tB2(x2y2))}eimin{βA1(x),βB2(x2y2)} ≤ min{tA1(x),min{tA2(x2),tA2(y2)}}eimin{βA1(x),min{βA2(x2),βA2(y2)}} min β (x),β (x ) , imin { A1 A2 2 }  min t (x),t (x ) ,  min β (x),β (y )  = min { A1 A2 2 } e { A1 A2 2 } min t (x),t (y )   (cid:26) { A1 A2 2 } (cid:27) = min{tA1×A2(x,x2),tA1×A2(x,y2)}eimin{βA1×A2(x,x2),βA1×A2(x,y2)}, uB1×B2((x,x2)(x,y2))eiγB1×B2((x,x2)(x,y2)) = max{uA1(x),uB2(x2y2))}eimax{γA1(x),γB2(x2y2)} ≤ max{uA1(x),max{uA2(x2),uA2(y2)}}eimax{γA1(x),max{γA2(x2),γA2(y2)}} max γ (x),γ (x ) , imax { A1 A2 2 }  max u (x),u (x ) ,  max γ (x),γ (y )  = max { A1 A2 2 } e { A1 A2 2 } max u (x),u (y )   (cid:26) { A1 A2 2 } (cid:27) = max{uA1×A2(x,x2),uA1×A2(x,y2)}eimax{γA1×A2(x,x2),γA1×A2(x,y2)}, z x y . ✷ 2 1 1 1 Similarly, we an prove it for ∈ X , and ∈ E G G cn 1 2 DGe(cid:28)nGiti=on(8. The omposit)ion ◦ of two -graphs is de(cid:28)ned as a pair 1 2 1 2 1 2 ◦ A ◦A , B ◦B , su h that: 1. sA1◦A2(x1,x2)eiαA1◦A2(x1,x2) = min{sA1(x1),sA2(x2))}eimin{αA1(x1),αA2(x2)}, Complex neutrosophi graphs 92 tA1◦A2(x1,x2)eiβA1◦A2(x1,x2) = min{tA1(x1),tA2(x2))}eimin{βA1(x1),βA2(x2)}, uA1◦A2(x1,x2)eiγA1◦A2(x1,x2) = max{uA1(x1),uA2(x2))}eimax{γA1(x1),γA2(x2)}, x x 1 2 for all , ∈ X, 2. sB1◦B2((x,x2)(x,y2))eiαB1◦B2((x,x2)(x,y2)) = min{sA1(x),sB2(x2y2))}eimin{αA1(x),αB2(x2y2)}, tB1◦B2((x,x2)(x,y2))eiβB1◦B2((x,x2)(x,y2)) = min{tA1(x),tB2(x2y2))}eimin{βA1(x),βB2(x2y2)}, uB1◦B2((x,x2)(x,y2))eiγB1◦B2((x,x2)(x,y2)) = max{uA1(x),uB2(x2y2))}eimax{γA1(x),γB2(x2y2)}, x x y 1 2 2 2 for all ∈ X , and ∈ E , 3. sB1◦B2((x1,z)(y1,z))eiαB1◦B2((x1,z)(y1,z)) = min{sB1(x1y1),sA2(z))}eimin{αB1(x1y1),αA2(z)}, tB1◦B2((x1,z)(y1,z))eiβB1◦B2((x1,z)(y1,z)) = min{tB1(x1y1),tA2(z))}eimin{βB1(x1y1),βA2(z)}, uB1◦B2((x1,z)(y1,z))eiγB1◦B2((x1,z)(y1,z)) = max{uB1(x1y1),uA2(z))}eimax{γB1(x1y1),γA2(z)}, z x y 2 1 1 1 for all ∈ X , and ∈ E . 4. sB1◦B2((x1,x2)(y1,y2))eiαB1◦B2((x1,x2)(y1,y2)) = min{sA2(x2),sA2(y2),sB1(x1y1)}eimin{αA2(x2),αA2(y2),αB1(x1y1)}, tB1◦B2((x1,x2)(y1,y2))eiβB1◦B2((x1,x2)(y1,y2)) = min{tA2(x2),tA2(y2),tB1(x1y1)}eimin{βA2(x2),βA2(y2),βB1(x1y1)}, uB1◦B2((x1,x2)(y1,y2))eiγB1◦B2((x1,x2)(y1,y2)) = max{uA2(x2),uA2(y2),uB1(x1y1)}eimax{γA2(x2),γA2(y2),γB1(x1y1)}, x ,y x = y x y 2 2 2 2 2 1 1 1 for all ∈ X , 6 and ∈ E . cn Example 3.GConGsider the two -graphs, as shown in Fig. 4. Then, their 1 2 omposition ◦ is shown in Fig. 5. cn cn Proposition 2. The omposition of two -graphs is a -graph. 93 Yaqoob, Akram cn G G 1 2 Fig. 4: -graphs and cn G G 1 2 Fig. 5: -graph ◦ Complex neutrosophi graphs 94 1 2 Proof: As in the previous proof, we verify the onditions for B ◦B only. We prove the 2nd ase: sB1◦B2((x,x2)(x,y2))eiαB1◦B2((x,x2)(x,y2)) = min{sA1(x),sB2(x2y2))}eimin{αA1(x),αB2(x2y2)} ≤ min{sA1(x),min{sA2(x2),sA2(y2)}}eimin{αA1(x),min{αA2(x2),αA2(y2)}} min α (x),α (x ) , imin { A1 A2 2 }  min s (x),s (x ) ,  min α (x),α (y )  = min { A1 A2 2 } e { A1 A2 2 } min s (x),s (y )   (cid:26) { A1 A2 2 } (cid:27) = min{sA1◦A2(x,x2),sA1◦A2(x,y2)}eimin{αA1◦A2(x,x2),αA1◦A2(x,y2)}, tB1◦B2((x,x2)(x,y2))eiβB1◦B2((x,x2)(x,y2)) = min{tA1(x),tB2(x2y2))}eimin{βA1(x),βB2(x2y2)} ≤ min{tA1(x),min{tA2(x2),tA2(y2)}}eimin{βA1(x),min{βA2(x2),βA2(y2)}} min β (x),β (x ) , imin { A1 A2 2 }  = min min{tA1(x),tA2(x2)}, e  min{βA1(x),βA2(y2)}  min t (x),t (y )   (cid:26) { A1 A2 2 } (cid:27) = min{tA1◦A2(x,x2),tA1◦A2(x,y2)}eimin{βA1◦A2(x,x2),βA1◦A2(x,y2)}, uB1◦B2((x,x2)(x,y2))eiγB1◦B2((x,x2)(x,y2)) = max{uA1(x),uB2(x2y2))}eimax{γA1(x),γB2(x2y2)} ≤ max{uA1(x),max{uA2(x2),uA2(y2)}}eimax{γA1(x),max{γA2(x2),γA2(y2)}} max γ (x),γ (x ) , imax { A1 A2 2 }  max u (x),u (x ) ,  max γ (x),γ (y )  = max { A1 A2 2 } e { A1 A2 2 } max u (x),u (y )   (cid:26) { A1 A2 2 } (cid:27) = max{uA1◦A2(x,x2),uA1◦A2(x,y2)}eimax{γA1◦A2(x,x2),γA1◦A2(x,y2)}, z x y . x y 2 1 1 1 2 2 2 Similarly, we an prove it for ∈ X , and ∈ E In the ase , ∈ X ,