PYTHAGOREAN FUZZY GRAPHS: SOME RESULTS

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1 YTHAGOREAN FUZZY GRAHS: SOME RESULTS Rajkumar Verma José M Merigó Manoj Sahni 3 Department of Management Control and Information Systems University of Chile Av Diagonal araguay 57 Santiago CHILE Department of Applied Sciences Delhi Technical Campus (Affiliated to Guru Govind Singh Indraprastha University Delhi) 8/ Knowledge ark-iii Greater Noida-0306 Uttar radesh INDIA 3 Department of Mathematics School of Technology andit Deendayal etroleum University Gandhinagar Gujarat INDIA rverma@fenuchilecl jmerigo@fenuchilecl manoj_sahani7@rediffmailcom ABSTRACT Graph theory has successfully used to solve a wide range of problems encountered in diverse fields such as medical sciences neural networks control theory transportation clustering analysis expert systems image capturing and network security In past few years a number of generalizations of graph theoretical concepts have developed to model the impreciseness and uncertainties in graphical network problems A ythagorean fuzzy set is a powerful tool for describing the vague concepts more precisely The ythagorean fuzzy set-based models provide more flexibility in handling the human judgment information as compared to other fuzzy models The objective of this paper is to apply the concept of ythagorean fuzzy sets to graph theory This work introduces the notion of ythagorean fuzzy graphs (FGs) and describes a number of methods for their construction We then define some basic operations on FGs and prove some of their important properties The work also discusses the notion of isomorphism between ythagorean fuzzy graphs with a numerical example Further we introduce the concept of the strong ythagorean fuzzy graph and the complete ythagorean fuzzy graph In addition the paper also proves some results on self-complementary self-weak

2 complementary with ythagorean fuzzy strong graphs and ythagorean fuzzy complete graphs Keywords: ythagorean fuzzy sets; fuzzy graphs; intuitionistic fuzzy sets Intuitionistic fuzzy graphs Introduction Zadeh [53] introduced the notion of fuzzy sets to model the uncertainty or vague concepts by assigning a membership degree corresponding to each element whose range is in between 0 and Since the pioneering work of Zadeh the fuzzy set theory has been used in different disciplines such as management sciences engineering mathematics social sciences statistics signal processing artificial intelligence automata theory medical and life sciences In 98 Atanassov [6] generalized the idea of fuzzy sets and introduced a new set theory called the intuitionistic fuzzy sets In the intuitionistic fuzzy set each element has degrees of membership and nonmembership whose sum lies between 0 and In last three decades the intuitionistic fuzzy set theory has been widely studied and a great number of applications have been developed in various fields including decision making [3 4 43] medical diagnosis [ ] market prediction [8] clustering analysis [ ] and pattern recognition [0- ] Graph theory is an important branch of applied mathematics and has numerous applications in different disciplines including computer science economics social sciences chemistry physics system analysis neural networks electrical engineering control theory transportation architecture and communication [4 7] However in many realistic situations some aspect of a graph-theoretic problem may be uncertain and cannot be represented by Euler s graph To handle such type of situations in 975 Rosenfeld [35] generalized the Euler s graph theory and proved the basic results on fuzzy graphs (FGs) Bhattacharya [7] made some comments on FGs and established some connectivity concepts regarding fuzzy cutnotes and fuzzy brides Further Bhutani [9] studied the automorphisms on fuzzy graphs and proved a number of properties connected with the complete fuzzy graph Mordeson and eng [8] defined some basic operations on fuzzy graphs Mcallister [6] discussed theoretical and computational aspects of fuzzy intersection graphs by matrix representation Bhattacharya and Suraweera [8] proposed an algorithm to find the connectivity of a pair of nodes in a fuzzy graph In 993 Mordeson [7] studied fuzzy line graph and proved the necessary and sufficient condition for a fuzzy graph to

3 be isomorphic to its corresponding line graph In 00 Mordeson and Nair [9] defined the complement of a fuzzy graph Later Sunitha and Kumar [4] proposed a modified definition of the complement of a fuzzy graph and proved some properties on self-complementary fuzzy graphs The cofuzzy graphs were studied by Akram [] in 0 Samanta and al [38] discussed the notion of fuzzy planar graphs and made a comparative study between Kuratowski s graphs and fuzzy planar graph In 994 Shannon and Atanassov [39] further generalized the fuzzy graph theory and proposed the intuitionistic fuzzy graphs (IFGs) with some fundamental results Karunambigai and arvathi [0] analyzed the properties of minmax intuitionistic fuzzy graphs arvathi et al [30] defined some operations on IFGs and studied their properties Karunambigai et al [] discussed the constant-ifgs and totally constant-ifgs In 0 Akram and Davvaz [4] studied the strong- IFGs and proved some of their properties Further Akram and Dudek [] proposed intuitionistic fuzzy hypergraphs and discussed their applications Alshehri and Akram [5] developed the notion of multigraphs planar graphs and dual graphs under intuitionistic fuzzy environment Sahoo and al [36] defined different types of product operations on IFGs Recently Sahoo and al [37] have proposed the idea of intuitionistic fuzzy tolerance graph intuitionistic fuzzy - tolerance graph and discussed their applications Due to their flexibility and applicability a number of researchers have been started work on intuitionistic fuzzy graph theory and proved a number of interesting results More recently Yager [50 5] has proposed the ythagorean fuzzy set (FS) as an effective tool for handling/modeling the uncertainty or vague information more adequately in real-world situations In FSs the sum of squares of the degrees of membership and nonmembership is less than or equal to For example if a decision maker provides the membership degree 06 and nonmembership degree 07 in his evaluation then this situation cannot handle by intuitionistic fuzzy set theory because of > However it is easily observed that < that is to say the ythagorean fuzzy set (FS) is capable to represent this evaluation information In other words the FSs are more powerful to handle problems in uncertain situations Under FS environment many researchers have started work in different directions and obtained various significant results Yager and Abbasov [5] established a relation between ythagorean membership degrees and complex numbers and proved that the ythagorean degrees are a 3

4 subclass of complex numbers Zhang and Xu [59] made a detailed study on ythagorean fuzzy sets and proposed an extension of the TOSIS method with ythagorean fuzzy information eng and Yang [3] defined the division and subtraction operations on FNs Reformat and Yager [34] studied the collaborative-based recommender systems under ythagorean fuzzy environment Dick et al [5] discussed some operations on ythagorean and complex fuzzy sets Ma and Xu [5] defined symmetric ythagorean fuzzy weighted geometric/ averaging operators and studied their applications in multicriteria decision making eng and Yang [3] developed MABAC method under ythagorean fuzzy environment Zhang [57] discussed multicriteria ythagorean fuzzy decision analysis using hierarchical UALIFLEX approach with the closeness index based ranking methods Zeng et al [55] proposed a hybrid method for solving ythagorean fuzzy multiple-criteria decision-making problems eng et al [33] introduced some ythagorean fuzzy information measures and discussed their applications Further Zeng [54] developed a new method to solve ythagorean fuzzy multiattribute group decision making problems In recent years many researchers [ ] have developed a number of aggregation operators to aggregate different FNs Intuitionistic fuzzy graphs have been successfully applied in solving many problems connected with different areas [3 9] But there are many problems in real life which cannot be represented adequately by intuitionistic fuzzy graphs So we need a more general graph theory to tackle such type of situations The aim of this work is to develop the graph-theoretic concepts under ythagorean fuzzy environment For doing so we first propose the notion of ythagorean fuzzy relation (FR) and ythagorean fuzzy graph (FG) as a further generalization of FGs and IFGs We then define some basic operations on FGs and prove some their important properties We also study the isomorphism and weak isomorphism between FGs Further the work proposes the idea of the strong ythagorean fuzzy graph and complete ythagorean fuzzy graph In addition we also prove some results on self-complementary and self-weak complementary with ythagorean fuzzy strong graphs and ythagorean fuzzy complete graphs This paper is organized as follows: Section describes some prerequisite material on graph theory intuitionistic fuzzy graph theory and ythagorean fuzzy sets Section 3 proposes the idea of ythagorean fuzzy relation ythagorean fuzzy graph and some basic operations including 4

5 Cartesian product composition union join and complement on ythagorean fuzzy graphs Some important properties of different operations on ythagorean fuzzy graphs also proved in this section with illustrative numerical examples In Section 4 we define an isomorphism between ythagorean fuzzy graphs Section 5 introduces the notion strong ythagorean fuzzy graph and complete ythagorean fuzzy graph Further a number of propositions are proved on strong ythagorean fuzzy graphs and complete ythagorean fuzzy graphs Section 6 concludes the paper with some future directions reliminaries This section presents a brief review of graph theory intuitionistic fuzzy graphs and ythagorean fuzzy sets which will be used for further development Some basic definitions in graph theory Definition (Graph) [6]: A graph G V E consists of two sets V and E The elements of V are called vertices and the elements E are called edges (formed by a pair of vertices) Two vertices u and v in G are said to be adjacent if uv is an edge ing Definition (Simple Graph) [6]: A graph is simple if it has no parallel edges or self-loops Definition 3 (Complete Graph) [6]: A complete graph is a simple graph in which every pair of distinct vertices is connected by an edge Definition 4 (Complementary Graph) [6]: The complement of a simple graph G is the graph G that has the same vertices as G such that any two vertices are adjacent in G if and only if they are not adjacent in G Definition 5 (Cartesian product) [6]: G The Cartesian product of two simple graphs V E and G V E is a graph defined by G G G V E with V V V E u u u v : u V u v E u v v v : vv u v E and Definition 6 (Composition) [6]: The composition of two simple graphs G V E G V E is a graph defined by G G V V E E E u u v v u v E u v and E is defined as in G G G G and where G G Note that 5

6 Definition 7 (Union) [6]: The union of two simple graphs G V E and G V E a simple graph defined by G G G V V E E is Definition 8 (Join) [6]: The join of two simple graphs G V E and G V E denoted by G G V V E E E is where E represents the set of all edges joining the nodes of V and V and assume that V V Definition 9 (Isomorphism) [6]: Two graphs G V E and G V E isomorphic written G G such that are if there are bijections :V G V G and : E G E G if and only if G s uv isomorphism between G andg Intuitionistic fuzzy graphs: Basic results G s u v Atanassov [6] extended the fuzzy set to the IFS and defined as follows: Such a pair of mapping is called Definition (Intuitionistic Fuzzy Set) [6]: An intuitionistic fuzzy set defined in a finite universe of discourse U u u u where Here the numbers n is given by u u u x U () : 0 : 0 and u U u and u non-membership of uu in u U 0 u u u U () respectively denote the degree of membership and degree of For each intuitionistic fuzzy set inu the intuitionistic fuzzy index (hesitation degree) can be defined as u u u u U (3) If u 0 u U then IFS becomes fuzzy set Therefore fuzzy sets are special cases of IFSs Shannon and Atanassov [39] defined intuitionistic fuzzy graphs as an extension of fuzzy graph theory 6

7 Definition (Intuitionistic Fuzzy Relation) [39]: Let S andt be two arbitrary finite nonempty sets An intuitionistic fuzzy relation (IFR) RS T intuitionistic fuzzy set of the form: in the universe S T is an R s t s t s t s S t T (4) where : 0 and : : 0 R s t S T and Here the numbers st and st of s ts T in the relation R R R R s t s t S T (5) R 0 s t s t s t S T (6) R R R denote the degrees of membership and non-membership Definition 3: Let and be two intuitionistic fuzzy sets defined inu Further suppose v is an intuitionistic fuzzy relation on U Then v is said to be an intuitionistic fuzzy relation on u v u v max u v U An IFR on U if u v min u v is called symmetric if u v v uand u v v u u v U Definition 4 (Intuitionistic Fuzzy Graph) [39]: An intuitionistic fuzzy graph (IFG) with underlying setv is a pairg and where is an IFS in V with 0 m m m V and is an IFS in E V V such that u v min u v and and u v max u v (7) 0 u v u v m n E (8) Here and represent the intuitionistic fuzzy vertex set of G and the intuitionistic fuzzy edge set ofg respectively Note that here is a symmetric intuitionistic fuzzy relation on Definition 5 (Strong Intuitionistic Fuzzy Graph) []: An IFG G strong intuitionistic fuzzy graph if min and u v u v is said to be a u v max u v uv E (9) 7

8 Definition 6 (Complete Intuitionistic Fuzzy Graph) []: An IFG G complete intuitionistic fuzzy graph if min and u v u v 3 ythagorean Fuzzy Set: Basic results is said to be a u v max u v u v V (0) The notion of ythagorean fuzzy sets (FSs) was introduced by Yager [50 5] in 03 to model the uncertain information in highly complex realistic problems that intuitionistic fuzzy sets cannot capture Definition 3 (ythagorean Fuzzy Set) [50 5]: A ythagorean fuzzy set defined in a finite universe of discourse U u u u where where the numbers n is given by x u u u U () : 0 : 0 and u U u and u u U represent the degree of membership and the degree of nonmembership respectively of uu in 0 u u uu () Additionally for each ythagorean fuzzy set in U the hesitation degree or the ythagorean index can be defined as u u u u U (3) Throughout this paper the set of all ythagorean fuzzy sets inu will be represented by FSU Definition 3 (Set Operations on ythagorean Fuzzy Set) [55]: Let FS U and u u u u U then some set operations can be defined as follows: i if and only if u u and ii if and only if and ; C iii u u u u U ; iv given by u u u u U (4) u u u U ; umax u u min u u u U ; 8

9 v umin u u max u u u U ; In the next section we propose the concept of ythagorean fuzzy relation (FR) and ythagorean fuzzy graphs (FGs) 3 ythagorean Fuzzy Graphs The ythagorean fuzzy graphs are a new generalization of Euler s graph theory that represents complex graphical problems more appropriately This approach has some advantages over intuitionistic fuzzy graphs in which the sum of the degrees of membership and nonmembership for any vertex or edge should be less than or equal to We start with the following formal definition of a ythagorean fuzzy relation Definition 3 (ythagorean Fuzzy Relation): Let S and T be two arbitrary finite non-empty sets A ythagorean fuzzy relation RS T of the form: in the universe S T is a ythagorean fuzzy set R s t s t s t s S t T (5) where : 0 and : 0 R s t S T and Here the numbers st and st of the ythagorean fuzzy relation R R R R s t S T (6) R 0 s t s t s t S T (7) R R R denote the degrees of membership and non-membership Definition 3: Let and be two ythagorean fuzzy sets defined inu Further suppose is a ythagorean fuzzy relation onu Then is called a FR on if u v min u v and A u v max B u B v u v U (8) Definition 33 (Symmetric ythagorean Fuzzy Relation): A ythagorean fuzzy relation R on U is called symmetric if R u v v u and u v v u R R u v U (9) R 9

10 Definition 34 (ythagorean Fuzzy Graph): A ythagorean fuzzy graph with the underlying set V is a pair G 0 where is a ythagorean fuzzy set in V with u u u V and is a ythagorean fuzzy set in E V V such that u v min u v and u v u v max (0) B 0 u v u v u v E () and Then and are the ythagorean fuzzy vertex set and the ythagorean fuzzy edge set ofg respectively Note that here is a symmetric ythagorean fuzzy relation on For convenience we use the notation uv for an element of E Throughout this paper G G and G represent respectively a crisp graph an intuitionistic fuzzy graph and a ythagorean fuzzy graph Remark 3: (i) When uv 0 and uv 0 between u and v for some u and v then there is no edge (ii) When either one of the following conditions is true then there is an edge between u and v : (a) uv 0 and uv 0 (b) uv 0 and uv 0 (c) uv 0 and uv 0 Example 3: Let G V E be a graph such that V a b c d and E ab bc cd da V V Let be a ythagorean fuzzy set in V and be a ythagorean fuzzy set in E defined by a0507 b0803 c0605 d0404 ab0407 bc05045 cd0305 da0406 Then the ordered pair G is a ythagorean fuzzy graph of G () 0

11 We now turn to study some basic operations on ythagorean fuzzy graphs: Definition 35 (Cartesian roduct of ythagorean Fuzzy Graphs): Let G G and be two ythagorean fuzzy graphs of the graphs G V E and G V E respectively where and are ythagorean fuzzy sets correspondingly defined in V and V ; and are ythagorean fuzzy sets in E V V and E V V respectively The Cartesian product of graphs G and as follows i u u min u u and G denoted byg G u u max u u u u V V V u u u v min u uv and ii is defined u u u v max u u v u V u v E u w v w min uv w and iii u w v w max u v w wv u v E

12 roposition 3: The Cartesian product of two ythagorean fuzzy graphs is a ythagorean fuzzy graph roof: Let t V uv E then we have t u t v min t uv t u v min min t u t v min min min t u t v min (3) t u t v max t uv t u v max max A t A u A t A v max max max t u t v max (4) Again let wv uv E then we have u wv w min uv w u v w min min u w v w min min min u w v w min (5) u wv w max uv w This completes the proof u v w max max u w v w max max max u w v w max (6)

13 Example 3: Consider G V E and G V E are two graphs such that V a b V c d E ab and E cd Let G and G graphs ing and G respectively where be ythagorean fuzzy a b ab (7) c d cd (8) It is easy to verify that G G is a FG of G G 3

14 Definition 36 (Composition of ythagorean Fuzzy Graphs): Let G G and be two ythagorean fuzzy graphs of the graphs G V E and G V E respectively where and are ythagorean fuzzy sets correspondingly defined in V and V ; and are ythagorean fuzzy sets in E V V and E V V respectively The composition of graphs G and follows i u u min u u and G denoted byg G u u max u u u u V V V t u t v min t uv and ii is defined as t u t u max t u v t V u v E u w v w min uv w and iii u w v w max u v w wv x y E u u v v min u v uv and iv 0 where E E u u v v u v E u v G G u u v v u v u v u u v v E E 0 max and E is defined as in the case for roposition 3: The composition of two ythagorean fuzzy graphs is a ythagorean fuzzy graph roof: Let t V uv E then we have t u t v min t uv t u v min min t u t v min min min t u t v min (9) 4

15 t u t v max t uv t u v max max t u t v max max max t u t v 5 max (30) Again let wv uv E then we have u wv w min uv w u v w min min u w v w min min min u w v w min (3) u wv w max uv w Further let 0 u v w max max u w v w max max max u w v w max (3) u u v v E E so u v E u v Then it follows that u u v v min u v uv u v u v min min u u v v min min min u u v v min (33) u u v v max u v uv u v u v max max u u v v max max min u u v v max (34)

16 This completes the proof Example 33: Let G V E and G V E be graphs such thatv a b V c d E ab and E cd Consider two ythagorean fuzzy graphs G G where a0603 b0507 ab0507 (35) and c d cd (36) By simple computation it is easy to verify that the graph G G is a FG of G G 6

17 Definition 37 (Union of ythagorean Fuzzy Graphs): Let G and G two ythagorean fuzzy graphs of the graphs G V E and G V E be respectively where and are ythagorean fuzzy sets correspondingly defined in V and V ; and are ythagorean fuzzy sets in E V V and E V V respectively The union of two ythagorean fuzzy graphs G andg denoted by G G is defined as follows i ii iii iv u u if u V V if u u u V V max if u u u u V V u A u if u V V if u A u u V V min if u A u A u u V V uv uv if uv E E if uv uv uv E E max if uv uv uv uv E E uv uv if uv E E uv if uv uv E E min if uv uv uv uv E E roposition 33: The union of two ythagorean fuzzy graphs is a ythagorean fuzzy graph roof: Let uv E E then we have uv max uv uv u v u v max min min u u v v min max max 7

18 u v min (37) uv min uv uv u v u v min max max u u v v max min min u v max (38) Again let uv E E then uv uv u min u v v min if u vv V u v v min max u v min if u V V vv V u u v v min max max u v min if u v V V (39) uv uv u max u v v max if u vv V u v v max min u v max if u V V vv V u u v v max min min u v max if x y V V (40) Similarly if uv E E then uv min u v and uv max u v This completes the proof 8 (4)

19 Example 34: Let G V E and G V E be graphs such thatv a b c d e a b c d f E ab bc be ce ad de and E ab bc bf bd cf V ythagorean fuzzy graphs G and G Consider two where a b c d e ab bc be ce de ad a b c d f ab bc bf bd cf Clearly G G is a ythagorean fuzzy graph of G G (4) (43) 9

20 Definition 38 (Join of ythagorean Fuzzy Graphs): Let G and G two ythagorean fuzzy graphs of the graphs G V E and G V E be respectively where and are ythagorean fuzzy sets correspondingly defined in V and V ; and are ythagorean fuzzy sets in E V V and E V V respectively The join of two ythagorean fuzzy graphs G and defined as follows i ii iii u u if u V V if u u u V V if u u u V V u u if u V V if u u u V V if u u u V V uv uv if uv E E if uv uv uv E E if uv uv uv E E G denoted byg G is 0

21 iv v uv uv if uv E E if uv uv uv E E if uv uv uv E E uv min u v max if uv u v uv E where Erepresents the set of all edges joining the nodes of V and V roposition 34: The join of two ythagorean fuzzy graphs is a ythagorean fuzzy graph roof: Let uv E then we have uv min u v u u v v min max max u v min uv max u v u u v v max min min u v max Again let uv E E the desired result directly obtains from roposition 33 (44) (45) This proves the proposition Example 35: Let G V E and G V E be graphs such that V a b V c d e E G ab and E cd de Consider two ythagorean fuzzy graphs G where a b ab (46) c0705 d0508 e0606 cd0508 de0507 (47) and

22 HereG G represents a ythagorean fuzzy graph of G G

23 roposition 35: Let G V E and G V E be crisp graphs and assume V V Further let and be ythagorean fuzzy sets defined on V V E and E respectively Then G G represents a ythagorean fuzzy graph of G G G if and only if G and G are ythagorean fuzzy graphs of G V E G V E respectively roof: Suppose that G G is a ythagorean fuzzy graph of G G Let uv E thenuv E and u vv V Thus uv uv min u v u v min uv uv max u v This shows that G u v and (48) max (49) G is also a ythagorean fuzzy graph This completes the proof roposition 36: Let G V E and G V E is a ythagorean fuzzy graph Similarly we can easily prove that be crisp graphs and assumev V Further let and be ythagorean fuzzy sets defined on V V E and E respectively Then G G G G is a ythagorean fuzzy graph of G GG if and only if and G are ythagorean fuzzy graphs of G V E V E respectively roof: The desired result follows from the proof of ropositions 34 and 35 Definition 39 (Complement of ythagorean Fuzzy Graph): The complement of a ythagorean fuzzy graph G follows (i) V V is a ythagorean fuzzy graph denoted by G and and defined as 3

24 (ii) u u (iii) (iv) u u uv u v uv u v uv uv min if 0 uv uv E min if 0 u v uv u v uv uv max if 0 xy uv E max if 0 Example 36: Let G be a ythagorean fuzzy where a0705 b0306 c080 d0504 (50) bc cd da 4

25 Note: One can easily verify that G G 4 Isomorphisms of ythagorean Fuzzy Graphs In this section we characterize various types of (weak) isomorphisms of ythagorean fuzzy graphs Definition 4 (Homomorphism of ythagorean Fuzzy Graphs): Let G and G be ythagorean fuzzy graphs A homomorphism g from G to G is a mapping g : V V which satisfies the following conditions: i u g u u g u u V ii u v g u f v u v g u f v u v E Definition 4 (Isomorphism of ythagorean Fuzzy Graphs): Let G G be ythagorean fuzzy graphs An isomorphism g from G to mapping g : V V which satisfies the following conditions: i u g u u g u u V ii u v g u f v u v g u f v u v E and G is a bijective 5

26 Example 37: Let G V E and G V E be graphs such thatv a a a a V b b b b E a a a a a a a a a a and E bb b b b b b b b b 3 4 two ythagorean fuzzy graphs G and G Consider where a0407 a0703 a30605 a40308 aa a3a 0504 aa40308 a4a0307 a3a4007 (5) b b b b 3 4 b b b b b b b b b b (5) A map g : V V g a b g a b g a b g a b Then we have defined by

27 a g a a g a a g a a g a a3 g a3 a3 g a3 a4 g a4 a4 g a4 aa3 g a g a3 aa3 g a g a3 a3a g a3 g a a3a g a3 g a aa4 g a g a4 aa4 g a g a4 a4a g a4 g a a4a g a4 g a a3a4 g a3 g a4 a3a4 g a3 g a4 Hence G is isomorphic to G Definition 43 (Weak Isomorphism of ythagorean Fuzzy Graphs): Let G and G be ythagorean fuzzy graphs A weak isomorphism g from G to G is a bijective mapping g : V V which holds the following properties: i g is homomorphism ii u g u u g u u V Example 38: Let G V E and G V E be graphs such that V a a V b b E G a a and E b b Consider two ythagorean fuzzy graphs G where and a a a a (53) b b bb (54) 7

28 Let g : V V be a mapping defined by g a b g a b Then we have a g a a g a a g a a g a p a a g a f a a a g a f a Hence the above mapping represents a weak isomorphism Definition 44 (Co-weak Isomorphism of ythagorean Fuzzy Graphs Let G and G be ythagorean fuzzy graphs A co-weak isomorphism g from G to G is a bijective mapping g : V V which holds the following properties: i g is homomorphism ii u v g u f v u v g u f v u v E Example 38: Let G V E and G V E be graphs such that V a a V b b E G a a and E b b Consider two ythagorean fuzzy graphs G where a a a a (55) b b bb (56) and 8

29 Let g : V V be a mapping defined by g a b g a b Then we get a g a a g a a g a a g a a a g a g a a a g a g a Hence the above mapping represents a co-weak isomorphism roposition 4: An isomorphism between ythagorean fuzzy graphs is an equivalence relation roof: Let G G and G 3 are three ythagorean fuzzy graphs For equivalence relation we will prove the reflexivity symmetry and transitivity for ythagorean fuzzy graphs i Reflexivity: It is obvious ii Symmetry: Let g : V V be an isomorphism of G ontog Then g is a bijective mapping defined by g u u u V and satisfying u g u u g u u V (57) u v g u g v u v g u g v u v E (58) Since g is a bijective mapping then it follows that g u u u V Thus g u u g u u u V (59) p g u g v u v g u g v u v u v E (60) 9

30 Hence a bijective mapping g : V V is an isomorphism G onto G Transitivity: Let g : V V and h : V V3 be the isomorphisms of G onto G and G onto G respectively Since a map g : V V 3 so we have defined by g u u u V is an isomorphism u g u u u g u u u V (6) Similarly a map h : V V3 u v g u g v u v u v g u g v v v uv E defined by h u u u V is an isomorphism so 3 u h u u u h u u u V (63) From (6) (63) and u v h u 3 h v u 3 v u v h u 3 h v u 3 v uv E g u u u V we have u g u u h u 3 h g u 3 u g u u h u 3 h g u 3 u V (6) (64) (65) From (6) and (64) we have uv g u g v uv hu 3 h v h g u 3 h g v uv g u g v uv h u 3 h v h g u 3 h g v uv E (66) Hence g f : V V3 is an isomorphism ofg ontog 3 This proves the result roposition 4: The weak isomorphism between ythagorean fuzzy graphs is a partial order relation roof: Let G G and G 3 are three ythagorean fuzzy graphs For partial order relation we will prove the reflexivity anti-symmetry and transitivity for ythagorean fuzzy graphs i Reflexivity: It is obvious 30

31 ii Anti-Symmetry: Let g : V Vbe a weak isomorphism of G ontog Then g is a bijective mapping defined by g u u u V and satisfying u g u u g u u V (67) p uv g u g v u v g u g v u v E Again let h : V V be a weak isomorphism of G onto G Then h is a bijective mapping defined by h x x x V and satisfying (68) u h u u h u u V (69) uv hu hv u v h u h v u v E The inequalities (68) and (70) hold on V and V only when ythagorean fuzzy graphs G G have the same numbers of edges and the corresponding edges have the same weight Hence G and G are identical Transitivity: Let g : V V and h : V V3 be the weak isomorphisms of G onto G and onto G 3 respectively Since a map g : V V isomorphism so we have defined by (70) and G g u u u V is a weak u g u u u g u u u V (7) Similarly a map h : V V3 u v g u f v u v u v g u f v v v uv E defined by h u u u V is a weak isomorphism so 3 u h u u u h u u u V (73) u v h u 3 g v u 3 v u v h u 3 g v u 3 v uv E (7) (74) 3

32 From (7) (73) and g u u u V we have u g u u h u 3 h g u 3 u g u u h u 3 h g u 3 u V From (7) and (74) we have uv g u g v uv hu 3 h v h g u 3 h g v uv g u g v uv h u 3 h v h g u 3 h g v uv E (75) (76) Hence h g : V V3 is a weak isomorphism of G onto G 3 This proves the roposition Definition 45: A ythagorean fuzzy graph G roposition 43: If G roof: Let G is called self-complementary if G G is a self-complementary ythagorean fuzzy graph then uv uv uv min u v (77) uv uv max u v (78) uv be a self-complementary FG Then there exist isomorphisms such that and g u u and and g u g v uv By the definition ofg we have g u u u V (79) g u g v uv u v V (80) min g u g v g u gv g u g v (8) ie min uv u v g u g v ie uv g u g v min u v uv uv uv ie uv min u v uv uv 3

33 (8) ie uv min u v Similarly we can show that This proves the proposition uv uv uv uv min u v (83) uv Remark: The conditions given in roposition 43 are not sufficient In the following example G is not isomorphic to G but uv uv uv uv min u v uv min u v uv 33

34 roposition 44: For any ythagorean fuzzy graph G then G is a self-complementary ythagorean fuzzy graph roof: Assume G ThenG min if uv min u v (84) max uv u v u vv (85) is a ythagorean fuzzy graph such that uv max u v u vv uv u v G under the identity mapping defined on V This proves the result Remark 4: The conditions given in roposition 44 are not necessary In the following example G G where the isomorphism f : V V is given by f a b f b d f c a f d c f e e but min uv u v uv max u v u vv 34

35 roposition 43: Let G (i) G G G G (ii) G G G G and G be two ythagorean fuzzy graphs Then roof: (i) Let I : V V V V be the identity map To prove (i) it is enough to prove that u u u u and uv uv uv uv From the definition of the complement of the ythagorean fuzzy graph we have u u 35

36 u if u V u if u V u if u V u if u V u u u u if u V u if u V u if u V u if u V u min uv u v uv min u v uv if uv E E min u v min u v if uv E min u v uv if uv E min u v uv if uv E min u v min u v if uv E uv if uv E uv if uv E 0 if uv E uv max uv u v uv max u v uv if uv E E max u v max u v if uv E (88) (86) (87) 36

37 max u v uv if uv E max u v uv if uv E max u v max u v if uv E uv if uv E uv if uv E 0 if uv E uv (89) This proves the result (ii) Let I : V V V V be the identity map To prove (ii) it is enough to prove that u u u u and uv uv uv uv From the definition of the complement of the ythagorean fuzzy graph we have u u u if u V u if u V u if u V u if u V u u u u u if u V u if u V u if u V u if u V u u (90) (9) 37

38 min uv u v uv min u v uv if uv E min u v uv if uv E min u 0 if v uv E uv if uv E uv if uv E min u if v uv E uv if uv E or E min u if v uv E uv max uv u v uv max u v uv if uv E max u v uv if uv E max u 0 if v uv E uv if uv E or E max u if v uv E uv This proves the result (9) (93) 5 Strong and complete ythagorean Fuzzy Graphs 5: Strong ythagorean Fuzzy Graphs Definition 5 ( -Strong ythagorean Fuzzy Graph): A FG G strong ythagorean fuzzy graph if uv min u v and uv u v and is called - max (94) 0 uv uv uv E (95) 38

39 Example: Definition 5 ( -Strong ythagorean Fuzzy Graph): A FG G is called -strong ythagorean fuzzy graph if uv min u p v and uv u v and Example: max (96) 0 uv uv uv E (97) 39

40 Definition 53 (Strong ythagorean Fuzzy Graph): A FG G ythagorean fuzzy graph if uv min u p v and uv u v and Example: is called strong max (98) 0 uv uv uv E (99) roposition 5: If G andg are strong-fgs theng G also strong-fgs G G and G G roof: These can easily prove on lines similar to proof of ropositions 3 3 and 34 Remark : The union of two strong-fgs is not necessarily a strong-fg Example: We consider the following example: are 40

41 roposition 5: If G G is a strong-fg then at least G or G must be strong roof: Assume G andg are not strong-fgs Then there exists uv E and uv E such that u v u v uv min u v min (00) u v u v u v max u v max (0) 4

42 Further assume that Let min u v u v u v u (0) E t u t v t V u v E u w v w wv u v E and consider t u t v E then we have and Therefore Hence t u t v min t uv min t u v (03) u u min u u and u v min u v (04) t u t v t u v min min (05) Similarly we can easily show that t u t v min t u t v (06) t u t v max t u t v (07) From (06) and (07) G G is not a strong-fg This is a contradiction This proves the roposition Remark: If G is strong and G is not strong theng G may or may not be strong Example 38: We consider the following examples: 4

43 43

44 roposition 53: If G G is a strong-fg then at least G or G must be strong roof: It can be proved in a similar way as roposition 5 roposition 54: If G G is a strong-fg then at least G or G must be strong roof: The proof is similar to roposition 5 44

45 Definition 54 (Complement of Strong ythagorean Fuzzy Graph): Let G strong-fg of G V E The complement of G G (i) V V (ii) u u (iii) (iv) and defined as follows u uv u uv u v uv uv u v uv be a is a strong-fg denoted by 0 if 0 uv uv E min if 0 0 if 0 uv uv E max if 0 Definition 55 (Self Complement of Strong ythagorean Fuzzy Graph): A strong-fg G is called self-complementary if G G Example: Consider a graph G V E such as V a b c and E ab bc Let G be a strong-fg of G where is a ythagorean fuzzy set in V and is a ythagorean fuzzy set in E defined by a b c ab bc (08) 45

46 Clearly G G hence G is a self complementary roposition 55: Let G be a strong-fg If uv min u v uv max u v u vv then G is self-complementary roof: Let G and be a strong ythagorean fuzzy graph such that uv min u v and I : V V gives G G roposition 56: Let G roof: Let G g : V V such that uv max u v u vv Identity mapping Hence G is self-complementary uv uv be a self-complementary strong-fg Then uv u v min (09) uv uv u v max (0) uv be a self-complementary strong-fg Then there exists an isomorphism and g u u and and g u g v uv Using the definition ofg we have g u u u V () g u g v uv u v V () 46

47 g u g v min g u g v ie uv min u v or uv u v Similarly we can show that This completes the proof uv uv min (3) uv uv u v max (4) uv roposition 57: Let G and G be two strong-fgs Then G G if and only if G G roof: Let G and G be isomorphic then there exists a bijective mapping g : V V u g u and and uv g u g v and By the Definition 54 we have and satisfying u g u u V (5) uv g u g v uv E (6) g u g v u min u y v min g u g v g u g v uv E u min u y v max g u g v From (7) and (8) we get G G The proof of converse part is straightforward (7) (8) roposition 58: Let G and G be two strong-fgs If there is a weak isomorphism between G and G then there exist a weak isomorphism between G and G roof: Let g be a weak isomorphism between G and G then g : V V mapping satisfying and g u u u V such that u g u and u g u u V is a bijective (9) u v g u g v and (0) u v g u g v uv E 47

48 Since g : V V is a bijective mapping then g u u u V Thus g : V V g u u and Using the Definition 54 we have u v min u v and min g u min g v u u uv is also bijective mapping such that () g u u u V () u v max u v Thus max g u max uv g : V V g v u u is a weak isomorphism between G and G (3) roposition 59: If there is a co-weak isomorphism between two strong-fgs G and G then there is a homomorphism between G and G roof: It can be proved similar to roposition 58 5: Complete ythagorean Fuzzy Graphs Definition 5 (Complete -Strong ythagorean Fuzzy Graph): A FG G is called a complete -strong ythagorean fuzzy graph if u v min u v and u v u v and max (4) 0 u v u v u vv (5) Definition 5 (Complete -Strong ythagorean Fuzzy Graph): A FG G is called a complete -strong ythagorean fuzzy graph if 48

49 u v min u v and u v u v and max (6) 0 u v u v u vv (7) Definition 53 (Complete ythagorean Fuzzy Graph): A FG G is called a complete ythagorean fuzzy graph if u v min u v and u v u v and Example: We consider the following examples: max (8) 0 u v u v u vv (9) 49

50 roposition 5: If G ythagorean fuzzy graph roof: It follows from roposition 3 is a complete-fg then G G ** ** is also a complete Definition 54 (Complement of Complete ythagorean Fuzzy Graph): Let G complete-fg ofg V E The complement of G graph denoted by G (v) V V (vi) u u (vii) (viii) u u xv and defined as follows uv u p v uv be a is a complete ythagorean fuzzy 0 if 0 uv u vv min if 0 uv u p v uv 0 if 0 uv u vv max if 0 Definition 55 (Self Complement of Complete ythagorean Fuzzy Graph): A complete-fg G is called self-complementary if G G 50

51 roposition 5: In a self-complementary complete ythagorean fuzzy graph G have uv uv u v we min (30) uv and uv u v uv max (3) uv roof: The proof is similar to roposition 55 roposition 53: Let G be a complete-fg of G V E uv min u v and If uv max u v u vv (3) then G is self-complementary roof: Let G Then G be a complete-fg such that uv min u v and G uv max u v u vv under the identity map therefore G is self-complementary roposition 54: Let G and G be complete-fgs Then G G if and only if G G roof: The proof is similar to roposition 57 6 Conclusions The work has extended the graph theoretical results under ythagorean fuzzy environment We have developed the concept of FGs as a generalization of fuzzy and intuitionistic fuzzy graphs ythagorean fuzzy graphical models provide more precision flexibility and compatibility to the user for describing the uncertainty in many combinatorial problems in different areas We have defined some basic operations such as the Cartesian product composition union join and complement on ythagorean fuzzy graphs and proved a number of their properties Further the work has developed the idea of isomorphism between ythagorean fuzzy graphs and illustrated with a numerical example We have also introduced the idea of strong ythagorean fuzzy graphs and complete ythagorean fuzzy graphs and proved some results with these graphs In future work we will study different types of graphs including line graphs hypergraphs cographs constant graphs cycles with ythagorean fuzzy information Different types of arcs in 5

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