Fuzzy Norm on Fuzzy Autocatalytic Set (FACS) of Fuzzy Graph Type-3

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1 ISSN Mathematics and Statistics Journal home page: (2): Fuzzy Norm on Fuzzy Autocatalytic Set (FACS) of Fuzzy Graph Type-3 1 Umilkeram Qasim Obaid and 2 Tahir Ahmad 1 Department of Mathematics, College of Science, AL-Mustansiriya University, Baghdad, Iraq. 2 Department of Mathematical Science and Centre for Sustainable Nano Materials, IbnuSina Institute for Scientific and Industrial Research, Universiti Teknologi Malaysia, Skudai, Johor, Malaysia. Address For Correspondence: Umilkeram Qasim Obaid, Department of Mathematics, College of Science, AL-Mustansiriya University, Baghdad, Iraq. umilkeram@yahoo.com ToCite ThisArticle:UmilkeramQasimObaid and 1Tahir Ahmad. Fuzzy Norm on Fuzzy Autocatalytic Set (FACS) of Fuzzy Graph Type-3. Mathematics and Statistics Journal, 3(1): 12-24, 2017 A B S T R A C T Fuzzy Autocatalytic Set (FACS) of fuzzy graph Type-3 particularly on the fuzziness of normed structure of FACS and its relation to cycles in FACS is explored. In other words, a fuzzy norm on a FT3-cycle space of FACS which will called a fuzzy normed cycle space of FACSis given and interpreted the catalytic chain reaction between two cycles in FT3-cycle space of FACS. We investigate some of its important properties such as the notions of convergence, Cauchyness and completeness infuzzy normed cycle space of FACS. Then, the application of the fuzzy normed cycle space of FACS to the clinical incineration process is elaborated. Keywords:FT3-cycle space of FACS, fuzzy autocatalytic set, fuzzy graph, incineration process, normed space of FACS INTRODUCTION Fuzzy theory has been studied for quite extensively in conjunction with problems in mathematics subjects including set theory, graph theory, measure theory, control theory, differential equations, topology and most recently functional analysis. The fuzzy characteristics that were mainly introduced by Zadeh in 1965 were merged into the crisp graph. In this regard, Rosenfeld (1975)unveiled his design for the basic structure of fuzzy graph. Ever since that, fuzzy graph has been promptly expanded and implemented in diverse fields. The study on a modeling of clinical waste incineration process in Malacca (Sabariah, 2005; Tahir et al., 2010) is a significant example of the implementation of fuzzy graph theory. The notion of an Autocatalytic Set (ACS) as initiated by Jain and Krishna (1998; 2003) has been integrated with graph theoretical concepts in Nevertheless, ACS was deficient in explaining the incineration process (Sabariahet al., 2009). This led to the development of the modeling work of the process and adapted the fuzzy logic as an integral part of the system. In other words, fuzzy graph had succeeded to apply the modeling of this process as presented in Figure 1.Six main variables specified in the process were modelled as vertices (nodes) and the catalytic relationships were presented as edges (links). Representing the process as fuzzy graph made amalgamation three notions which are graph, autocatalytic set and fuzzy. This study was focused on fuzzy graph of type-3 and innovated a fresh notion called Fuzzy Autocatalytic Set (FACS) of fuzzy graph type-3. It was shown that the modeling fuzzy work (FACS) of the incineration process was more precise and suitable in describing the dynamics of this process (Tahir et al., 2010). Published Online in 19 May 2017 Open Access Journal Published BY IWNEST Publication 2017 IWNEST Publisher All rights reserved This work is licensed under the Creative Commons Attribution International License (CCBY).

2 13 UmilkeramQasimObaid and Tahir Ahmad, 2017 V 1 : Waste (particularly clinical waste) V 2 : Fuel V 3 : Oxygen V 4 : Carbon Dioxide V 5 : Carbon Monoxide V 6 : Other Gases including Water Fig. 1: FACS of fuzzy graph Type-3 for the incineration process (Tahir etal., 2010) The structure of FACS with its relation to normed space and its cycles was taken into consideration in more investigation on FACS of fuzzy graph Type-3 and the richness of directed graph through functional analysis advantages (Umilkeram and Tahir, 2016). More precisely, a new concept namely normed space of FACS of fuzzy graph Type-3 was defined and implemented in the modelling of the incineration process. This paper focuses on the representation of FACS in fuzziness of normed structure of FACS by the construction a fuzzy normed cycle space of FACS that will be presented in this paper. In fact, a notion of a fuzzy norm in a fuzzy graph theory setting is established and interpreted the catalytic chain reaction between two cycles in this structure of FACS and employed in the incineration process that have different catalytic chain reaction because of its fuzzy norm structure. In addition, some important results would be proven involving convergence, Cauchyness and completeness of this structure of FACS. Preliminaries: Some of the basic concepts and results that included definitions and theorems pertaining to this research are introduced. The theoretical foundations of FACS are also presented.a directed graph is a certain trends of a (undirected) graph which the lines (edges) are labeled from points (vertices) to other as shown in Figure 2. Henceforward, this kind of graph will be studied throughout this study. (a) (b) Fig. 2:Examples of Graphs. (a) Directed (b) Undirected A directed graph is introduced by a relation on a set where denotes the set of vertices and denotes the set of its edges (Epp, 1993). A directed graph is also called a crisp graph if all the values of edges are 1 or 0 and it is called a fuzzy graph if its values is between 0 and 1. In other words, a fuzzy graph is with a vertex set as the underlying set with a pair of functions such that and is a fuzzy relation on with for all and denotes the minimum of and (Rosenfeld, 1975). A path :,,,...,, from a vertex to a vertex in a fuzzy graph if its sequence of distinct vertices and edges starting from and ending at such that the membership value for. If and coincide in a path then we call as a cycle. Note that the underlying crisp graph of the fuzzy graph is referred to.

3 14 UmilkeramQasimObaid and Tahir Ahmad, 2017 Moreover, Blue et al. (1997; 2002) have given five interpretations of the fuzziness of graph that led to study the choice of one of its type to best describe the model of incineration process. Thereby, Tahir et al. (2010) found that Type-3 is to be the most convenient for the assignment of improving the model of the process as explained in (Sabariahet al., 2009). The Type-3 of fuzzy graph is defined as having crisp vertices and edges with fuzzy edge connectivity of the edges i.e. the edge has fuzzy head and tail. With this consideration, the following definition was adopted in obtaining the fuzzy graph representing the clinical incineration process. The formal definition of FACS is given as follows: Fuzzy autocatalytic set (FACS) is a subgraph where each of whose nodes has at least one incoming link with membership value (Tahir et al., 2010). Aforementioned, Figure 1 is a fuzzy graphical for the modelof the incineration process with 6 variables (namely waste ( ), fuel ( ), oxygen ( ), carbon dioxide ( ), carbon monoxide ( ), and other gases including water ( )) and 15 edges which based on the catalytic relationship between the variables. Thus, is the set of vertices and is the set of edges where for and. Hence, fifteen edges are the connection between these variables in the process and the membership values of each fuzzy connectivity of edge are given as follows:,,,,,,,,,,,,,., In this context, a cycle in a fuzzy graph is a directed closed path of its vertices such that each edge and each vertex (except starting point) is visited only once. Then, FACS of fuzzy graph Type-3 particularly on the structures of normed spaces and its relations to cycles in FACS has been studied by Umilkeram and Tahir (2016). Several new notions namely FT3-fuzzy detour cycles of FACS, a FT3-cycle space of FACS as a vector space, and normed space of FACS of fuzzy graph Type-3 were presented as follows. Definition 1: Let be a FACS of fuzzy graph Type-3. The FT3-cycle is a closed path of distinct vertices,,..., (except = ) such that the membership value,, for each fuzzy edge connectivity of FACS and n is the number of edges in this cycle. The length of FT3-cycle in FACS is calculated by = such that each edge in this cycle is traversed in the right direction (see Figure 3). Definition 2: Let n be the number of all edges in the FT3-cycle containing an edge in FACS. A length of FT3-cycle containing the edge with n, denoted by, in FACS of fuzzy graph Type-3 is defined as the maximum length of any FT3-cycle that goes through the edge with the number of all edges in these FT3-cycles is the same and equal to n. A FT3-cycle containing the edge of length is called FT3-fuzzy detour cycle such that. Fig. 3: FT3-fuzzy detour cycle in FACS

4 15 UmilkeramQasimObaid and Tahir Ahmad, 2017 Definition 3: For a graph, let denote the set of all FT3-fuzzy detour cycles and edge-disjoint unions of these cycles in FACS, and an empty graph which means all vertices separate from each other. Then, is called the FT3-cycle space of. Thereafter,Umilkeram and Tahir (2016) proved that under the sum operation and multiplicationoperation (see Theorem 1) forms a vector space over {0, 1} (with addition and multiplication modulo 2), where the vector addition and the scalar multiplication are given by (1) and for each, (2) Theorem 1: The FT3-cycle space of,, is a vector space over {0, 1} (with addition and multiplication modulo 2). Then, the normed space of FACSwas constructed using the notion of the FT3-cycle space of FACS as a vector space over together with the following norm. = max (3) This shows that the concept of the norm of a vector in the FT3-cycle space of, is a generalization of the concept of length in. In other words, a FT3-cycle space of forms a normed space with the function as presented by Eq. (3). Then, a new type of normed space (see Theorem 2) which is the normed space of FACS of fuzzy graph Type-3 was presented. Theorem 2: Let be a FT3-cycle space of over and = max be a real-valued function on. Then, is a normed space. Furthermore, the basic FT3-fuzzy detour cycles for each edge in FACS of the incineration processwith respect to this norm are determined in this process represented by the FT3-cycle space associated with a graph of FACS of the incineration process. Consequently, the basic FT3-fuzzy detour cycles with respect to the norm as presented in Eq. (3) when interpret physically means that each fuzzy connectivity of edge in a FT3-fuzzy detour cycle has a certain proportion of the chemical interaction with other edges to the greatest extent norm. Now, we look at the idea of fuzzy norm on a FT3-cycle space of and its relation with the crisp norm which is presented in Theorem 2. The novelty of this notion is the incorporation of a couple important definitions in our work that are fuzzy graph represented by FACS of fuzzy graph Type-3 and fuzziness of norm represented by fuzzy norm in our sense which is close to the adaptation of Bag and Samanta type (2013). However, the modern notion is defined in such a way that the corresponding metric fuzziness is fuzzy quasi (pseudo) FT3-metric of FACS as shown inumilkeram and Tahir (2014). RESULTS AND DISCUSSION Fuzzy Norm on a FT3-Cycle Space of FACS: In this section, a concept of fuzzy norm is given on a FT3-cycle space of which defined in Definition 3. It begins with taking a function defined on as given in Eq. (4), followed by discussion on its properties. We see how this equation can be specialized for the idea of fuzzy norm on a FT3-cycle space as follows. It is noted that the above function has the following properties given in the following remarks. Remark 1: It is clear that for all. Clearly, for all. The following statement is not satisfied that (for all if and only if ) for Eq. (4), due to the second condition of the norm of (see Theorem 2). i.e. if and only if the cycle and which means the number of all edges in this cycle is. This is a contradiction since. (4)

5 16 UmilkeramQasimObaid and Tahir Ahmad, 2017 if Evidently, for all and, if (i.e. if ). is nondecreasing function on. If and, then it is obvious that. Now,, then = = Hence, clearly,. If. In what follows, it could be also shown that idea of fuzzy norm on a FT3-cycle space as presented in Eq. (4) fulfills the main property of the in the following theorem. Theorem 3: Suppose is a FT3-cycle space of over and is a norm on. Then, defined in Eq. (4) satisfies the following relation: for all, and. Proof: The relation is proved as follows: If (1) with ; ; (2) (3) with ;, then it is easily verified the above relation. If (4) with, then by the function as presented in Eq. (4) and the fourth condition of the norm of, we have (5) Now, note that = = = = (6) since Hence, by Eq.(5) and Eq.(6), we have the relation and this concludes the proof. Thus, at this placement, combining these properties with known notions on fuzzy norm in functional analysis, especially Bag and Samanta type (2013), this brings to the notion of a fuzzy normed cycle space of FACS as given in the following. Definition 4: Let be a FT3-cycle space of over and be a norm on. Suppose is a fuzzy set defined by, Wheretis the number of all edges in the FT3-fuzzy detour cycle and a continuous t-norm is theusual multiplication for all and is satisfied the following properties, for all.

6 17 UmilkeramQasimObaid and Tahir Ahmad, for all, 2. for all, 3. for all,, 4. is a nondecreasing function on 5.,. Then, an ordered pair (, ) is said to be fuzzy normed cycle space of FACS and is called a fuzzy norm induced by a norm on, see Figure 4. all edges in the FT3-fuzzy detour cycle, where is the number of in FACS. and = max, where is the number of vertices is one of these FT3-fuzzy detour cycle in FACS. wheretis the number of all edges in the FT3-fuzzy detour cycle in FACS. If satisfies the mentioned conditions 1,2,3,4,5 in Definition 4 then isfuzzy norm induced by norm on FACS. Fig. 4: Graphical representation ofafuzzy norm induced by a norm on FACS It is to be noted here thata fuzzy norm on in the sense that relates the greatest extent norm to andtis the number of all edges in this cycle. Moreover, an important observation in Definition 4 that the function is considered as the degree of nearness

7 18 UmilkeramQasimObaid and Tahir Ahmad, 2017 between two cycles in. It means this function can be interpreted the catalytic chain reaction between two cycles infuzzy normed cycle space of FACS. Therefore, one can talk about a sequence of cycles in fuzzy normed cycle space of FACS and discuss the concepts of a convergent sequence and Cauchy sequence in (, ) in the following subsection. Convergence in Fuzzy Normed Cycle Space of FACS: Suppose that is FT3-fuzzy detour cycle in, then from Definition 4, has a fuzzy norm such that the cycle containing the edge with which is the number of all edges in this cycle. Then, take another cycle containing the edge with which is the number of all edges in this cycle and hence, has a fuzzy norm. Consequently, the degree of nearness between two cycles of the catalytic chain reaction between these cycles in a fuzzy normed cycle space of FACS (, ) is. Now, by the same argument, another cycle containing the edge with has a fuzzy norm, then the degree of nearness between two cycles of the catalytic chain reaction between these cycles in (, ) is. Continuing in this way, we obtain a sequence of FT3-fuzzy detour cycles in (, ) which related to a fixed cycle with a fuzzy norm as shown in Figure 5. Fig. 5: Graphical illustration of a sequence (, ) of FACS of FT3-fuzzy detour cycles in a fuzzy normed cycle space Without loss of generality, it can replace by means of due to each element in is its own negative (i.e. as usual writing, see Theorem 1). In other words, represents inverse of the sum operation of symmetric difference as given by Eq.(1) in the vector space of FT3-cycle space. Then, one can provide a notion of a convergent sequence of FT3-fuzzy detour cycles in (, ) as follows. Definition 5: Let (, ) be a fuzzy normed cycle space of FACS and is a fuzzy norm induced by a norm on. Then, a sequence of FT3-fuzzy detour cycles in converges to (i.e.

8 19 UmilkeramQasimObaid and Tahir Ahmad, 2017 ) if satisfying a condition that for all. One calls a convergent point ofa sequence, see Figure 5. The above definition can be illustrated with the values of a function for the sequence of the above cycles in which converges to other cycle. Thus, we could studythe degree of nearness between two cycles of the catalytic chain reaction between these cycles in a fuzzy normed cycle space of FACS. In other words, we could study the relationship between these FT3-fuzzy detour cycles in FACS by a notion of a convergent sequence for all cycles in. Theorem 4: The convergent point of a sequence is unique. Proof: Let and and assume that. Then, by Definition 5, and for all. Thus, we obtain (by ) (by condition 3of Definition 4) =. Then, by taking limit, imply that Thus,. By condition 2 of Definition 4, yield that and this is a contradiction. Therefore, and this complete the proof. Now, the notion of a Cauchy sequence can be given when we have many FT3-fuzzy detour cycles in and it is complicated to find the catalytic chain reaction between these cycles that related to a fixed cycle in (, ). Definition 6: Suppose (, ) is a fuzzy normed cycle space of FACS and is a fuzzy norm induced by a norm on. Then, a sequence of FT3-fuzzy detour cycles in is a Cauchy sequence if satisfying a condition that for all and (see Figure 6).

9 20 UmilkeramQasimObaid and Tahir Ahmad, Fig. 6: Graphical illustration of a Cauchy sequence space (, ) of FACS of FT3-fuzzy detour cycles in a fuzzy normed cycle Theorem 5: Let be a convergent sequence in a fuzzy normed cycle space of FACS (, ). Then, is also a Cauchy sequence. Proof: Since is a convergent sequence in (, ), then bydefinition 5, for all and is convergent point of a sequence. Thus, by ( ) and for all and, we obtain Now, it is clear that each subsequence of and by taking limit, imply that By condition 2 of Definition 4, yield that = converges to the same convergent point of a sequence for all and.. Hence, from Definition 6, is a Cauchy sequence in (, ). Common Characteristics of Fuzzy Norm and Norm on FT3-Cycle Space of FACS: Consider the FT3-cycle space over of FACS and the norm defined on by. Thus, from Theorem 2,

10 21 UmilkeramQasimObaid and Tahir Ahmad, 2017 (, ) is a normed space. Also, consider the fuzzy norm induced by a norm on and defined by, Thus, from Definition 4, (, ) is a fuzzy normed cycle space of FACS. In this section, some properties of the norm on a FT3-cycle space can be participated with the fuzzy norm. The question that comes to our mind is to know which of these characteristics can be generalized to this fuzzy norm. The following two theorems try to answer the question. Theorem 6: Let(, ) be a fuzzy normed cycle space of FACS and is a fuzzy norm deduced of the norm on. Then, is a Cauchy sequence in (, ) if and only if is a Cauchy sequence in (, ). Proof: Suppose is a Cauchy sequence ina normed space (, ). It means the sequence of FT3-fuzzy detour cycles in satisfies a condition, as usual, for all (7) Then, for all imply that. By taking limit, imply that Therefore, is a Cauchy sequence in (, ). = = 1 (by (7)) Conversely, let be a Cauchy sequence in (, ). Then, for all and. Thus, we have = 1. Hence, is a Cauchy sequence in (, ). Theorem 7: Let (, ) be a fuzzy normed cycle space of FACS and is a fuzzy norm deduced of the norm on. Then, is a convergent sequence in (, ) if and only if is a convergent sequence in (, ). Proof: Let is a convergent sequence in (, ). From Definition 5, for all, imply that = 1 = 1 cycles in is converges to in a normed space (, ). in (, ). i.e. the sequence of FT3-fuzzy detour Thus, from previous two theorems, it is noticed that if there exists a normed space (, ) that is not complete, then a fuzzy normed cycle space of FACS (, ) is also not complete that the fuzzy norm induced by the crisp norm. Implementation: Implementation of Fuzzy Normed Cycles Space of FACS to Clinical Incineration Process: The idea of a fuzzy normed cycle space of FACS is used to give a fuzzy norm associated with a graph of FACS for the incineration process. Thus, it is easily verified that the function in Eq. (4) is satisfied the five conditions in Definition 4 with a graph of FACS for the incineration process.then, the fuzzy norm of each FT3-fuzzy detour cycle in Figure 3 with respect to t which is the number of all edges in this cycle is computed as follows.

11 22 UmilkeramQasimObaid and Tahir Ahmad, The FT3-fuzzy detour cyclethat contained the edge is with norm, then 2. The FT3-fuzzy detour cyclethat contained the edge is with norm 3. The FT3-fuzzy detour cycle that contained the edge is withnorm 4. The FT3-fuzzy detour cycle that contained the edge is with norm 5. The FT3-fuzzy detour cycle that contained the edge is with norm 6. The FT3-fuzzy detour cycle that contained the edge is with norm 7. The FT3-fuzzy detour cycle that contained the edge is with norm, then 8. The FT3-fuzzy detour cycle that contained the edge is with norm 9. The FT3-fuzzy detour cycle that contained the edge is with norm 10. The FT3-fuzzy detour cycle that contained the edge is with norm 11. The FT3-fuzzy detour cycle that contained the edge is with norm 12. The FT3-fuzzy detour cycle that contained the edge is with norm 13. The FT3-fuzzy detour cycles that contained the edge is with norm, then 14. The FT3-fuzzy detour cycles that contained the edge is with norm, then 15. The FT3-fuzzy detour cycles that contained the edge is with norm, then Thus, each FT3-fuzzy detour cycle in fuzzy normed cycle space of FACS for the incineration process have different catalytic chain reaction because of its fuzzy norm structure. Besides that, a chain reaction means here a chemical reaction in which the products themselves causes additional reactions to take place and enhance the reaction which under specific conditions may quicken. However, the function is considered as the degree of nearness between two cycles of the catalytic chain of chemical reaction between these cycles in fuzzy normed cycle space of FACS for the incineration process as explained in the following subsection. It was shown in Umilkeram and Tahir (2016) that the basic FT3-fuzzy detour cycles with respect to the norm given in Eq. (3) are ten cycles due to the fact that and Then, we attempt to explain the function as the catalytic chain reaction between two cycles in fuzzy normed cycle space of FACS for the incineration process in the following details. From third condition in Definition 4, note that for all,,. Then, we consider a FT3- fuzzy detour cycle that contained the edge with which is the number of all edges in this cycle and observe that and

12 23 UmilkeramQasimObaid and Tahir Ahmad, 2017 =. and =. Thus, the catalytic chain of chemical reaction between two cycles is more nearness between two cycles among other cycles that associated to and which is equal to =. Now, by the same argument, one can take another cycle that contained the edge with which is the number of all edges in this cycle and hence, has a fuzzy norm. Consequently, the degree of nearness between two cycles of the catalytic chain reaction between these cycles in a fuzzy normed cycle space of FACS is and is computed as bellow. Thus, the catalytic chain of chemical reaction between two cycles is more nearness between two cycles among other cycles that associated to and which is equal to =. Now, by taking another cycle that contained the edge with which is the number of all edges in this cycle and hence, has a fuzzy norm. Then, the degree of nearness between two cycles of the catalytic chain reaction between these cycles in a fuzzy normed cycle space of FACS is and is computed as bellow.

13 24 UmilkeramQasimObaid and Tahir Ahmad, 2017 Thus, the catalytic chain of chemical reaction between two cycles is more nearness between two cycles among other cycles that associated to and which is equal to =. Continuing in this way, we obtain a sequence of the degree of nearness between two cycles of the catalytic chain reaction between these cycles in a fuzzy normed cycle space of FACSfor the incineration process. Conclusion: In this paper, we introduce a notion of a fuzzy norm in a graph theory setting, namely, a fuzzy normed cycle space of FACS. Indeed, constructing fuzzy norm based on the FT3-cycle space of FACS is established and interpreted the catalytic chain reaction between two cycles in thisspace. Consequently, each FT3-fuzzy detour cycle in fuzzy normed cycle space of FACS for the incineration process have different catalytic chain reaction because of its fuzzy norm structure.in addition, some important results have been proven involving convergence, Cauchyness and completeness of the fuzzy normed cycle space of FACS. REFERENCES Blue, M., B. Bush and J. Puckett, Applications of Fuzzy Logic to Graph Theory. Los Alamos National Laboratory, LA-UR Blue, M., B. Bush, and J. Puckett, Unified Approach to Fuzzy Graph Problems. Fuzzy Sets and Systems, 125(3): Epp, S.S., Discrete Mathematics with Applications. Boston: PWS Publishing Company. Jain, S. and S. Krishna, Autocatalytic Sets and Growth of Complexity in an Evolutionary Model. Physical Review Letters, 81(25): Jain, S. and S. Krishna, Graph Theory and the Evolution of Autocatalytic Networks. In: Handbook of Graphs and Networks. Bornholdt, S. and H. G. Schuster (Eds.). Berlin: John Wily and VCH Publishers, pp: Rosenfeld, A., Fuzzy Graphs. In: Fuzzy Sets and Their Applications to Cognitive and Decision Processes. Zadeh, L. A., K.S. Fu and M. Shimura (Eds.). New York: Academic Press. Sabariah, B., Thinking Mathematical: Modeling of a Clinical Waste Incineration Process using Novel Fuzzy Autocatalytic Set, PhD. Thesis, UniversitiTeknologi Malaysia-Johore, Malaysia. Sabariah, B., A. Tahir, and M. Rashid, Relationship between Fuzzy Edge Connectivity and the Variables in Clinical Waste Incineration Process. MATEMATIKA, 25(1): Tahir, A., B. Sabariah and A.A. Khairil, Modeling a Clinical Incineration Process Using Fuzzy Autocatalytic Set. Journal of Mathematical Chemistry, 47(4): Umilkeram, Q.O. and A. Tahir, Fuzzy quasi-metric spaces of Fuzzy Autocatalytic Set (FACS) of fuzzy graph Type-3. Life Science Journal, 11(9): Umilkeram, Q.O. and A. Tahir, Novel Structures of Normed Spaces with Fuzzy Autocatalytic Set (FACS) of FuzzyGraph Type-3. Research Journal of Applied Sciences, Engineering and Technology, 12(5): Zadeh, L.A., Fuzzy sets. Information and Control, 8: