Ratio Mathematica Volume 43, 2022 Group Decision Making in Conditions of Uncertainty using Fermat’s Weak Fuzzy Graphs and Beal’s Weak Fuzzy Graphs T M Nishad* B.Mohamed Harif† A.Prasanna‡ Abstract Decision making is a process of solving problems for choosing the best alternative. The best way to illustrate the alternatives and relation between them is a graph. Developing a fuzzy graph is the convenient way of illutration if there is uncertainty in alternatives or in their relation. In group decision making problems, according to a group of experts, the relation between alternatives involves measure of preference and non preference. Intuitionistic fuzzy graph has limitations to model such problems. In n- Pythagorean fuzzy graphs the hesitancy degree and other decision tools are restricted to second degree.To overcome the flaws of intuitionistic fuzzy graphs and n- Pythagorean fuzzy graphs, we introduced Fermat’s Fuzzy Graphs in 2022. In this paper the decision tools are generalized for Fermat’s Fuzzy Graphs. A practical example of selection of investement scheme is illustrated. Finally, Beal’s Fuzzy graphs is developed as generalization of Fermat’s Fuzzy Graphs. Keywords: Fuzzy Graph; Weak Fuzzy Graph; Fermat’s Fuzzy Graph; Fermat’s Weak Fuzzy Graphs; Beal’s Fuzzy Graph; Beal’s Weak Fuzzy Graphs. 2010 AMS subject classification: 03E72,05C72. *Research Scholar, Department of Mathematics, Rajah Serfoji Government College (Autonomous), (Affiliated to Bharathidasan University),Thanjavur-613005,Tamilnadu, India.; Email :nishadtmphd@gmail.com. †Assistant Professor and Research Supervisor, Department of Mathematics, Rajah Serfoji Government College (Autonomous),(Affiliated to Bharathidasan University),Thanjavur-613005, Tamilnadu, India.; Email:bmharif@rsgc.ac.in. ‡Assistant Professor, P G and Research Department of Mathematics, Jamal Mohamed College (Autonomous),(Affiliated to Bharathidasan University), Tiruchirappalli,-620020, Tamilnadu, India.; Email: apj_jmc@yahoo.co.in. ‡Received on September 15, 2022. Accepted on December15, 2022. Published on December30, 2022. DOI:10.23755/rm.v39i0.854. ISSN: 1592-7415. eISSN: 2282-8214. © Nishad et al. This paper is published under the CC-BY licence agreement. 208 mailto:nishadtmphd@gmail.com mailto:bmharif@rsgc.ac.in mailto:apj_jmc@yahoo.co.in T.M.Nishad, B.Mohamed Harif and A Prasanna 1. Introduction L A Zadeh in 1965[1] introduced fuzzy sets to describe the vagueness phenomena in real world problems.In 1975[2], Azriel Rosenfeld introduced fuzzy graphs. A.Prasanna and T M Nishad introduced weak fuzzy graphs in 2021 [3]. In 2009, Hongmei and Lianhua defined Interval Valued Fuzzy Graph (IVFG) [4] and in 2013 Talebi and Rashmanlou studied properties of isomorphism and complement of an IVFG[5]. To overcome the flaws of Intuitionistic Fuzzy Graph in simulation, Muhammed Akram, Amna Habib, etc. discussed specific types of Pythagorean Fuzzy Graphs and applications to decision making in 2018[6]. Fermat’s Weak Fuzzy graph and hesitancy degree in general scale are discussed by B.M Harif and T.M Nishad in 2022[7]. The American Banker and amateur mathematician Mr.Daniel Andrew Beal formulated the Beal’s conjecture in1993 [8] as a generalization of Fermat’s Conjecture. The contents of this article are as follows. In section 2 some fundamental concepts of Fuzzy Graphs and Fermat’s Fuzzy Graphs are reviewed.Section 3 illustrates the mathematical model of a group decision making problem using Fermat’s weak Fuzy Graph. Section 4 describes the generalized decision tools for Fermat’s Fuzzy Graphs. A practical example of selection of investment scheme is illustrated in section 5. In section 6 , the fundamental concepts of Beal’s Fuzzy Graph and some theorems are developed.The whole article is concluded in section 7. 2.Some Fundamental concepts of Fuzzy Graphs A mapping : [0,1]m A → from a non empty set A is a fuzzy subset of A. A fuzzy relation r on the fuzzy subset m , is a fuzzy subset of A A . A is assumed as finite non empty set. Definition 2.1: Suppose A is the underlying set. A fuzzy graph is a pair of functions G : (𝑚, 𝑟) where fuzzy subset : [0,1]m A → , the fuzzy relation r on 𝑚 is denoted by 𝑟 : A A →[0,1], such that for all ,u v A , we have 𝑟(𝑢, 𝑣) ≤ 𝑚(𝑢) ∧ 𝑚(𝑣) where stands minimum. G*: (𝑚∗, 𝑟∗) denotes the underlying crisp graph of a fuzzy graph G : (𝑚, 𝑟) where * { / ( ) 0}m u A m u=   and * {( , ) / ( , ) 0}r u v A A r u v=    . The nodes u and v are known as neighbours if 𝑟(𝑢, 𝑣) > 0. Definition 2.2.: A fuzzy graph G:(𝑚, 𝑟) is a strong fuzzy graph if 𝑟(𝑢, 𝑣) = 𝑚(𝑢) ∧ 𝑚(𝑣),∀(𝑢, 𝑣) ∈ 𝑟∗ . Definition 2.3: A fuzzy graph G :(𝑚, 𝑟) is a weak fuzzy graph if 𝑟(𝑢, 𝑣) < 𝑚(𝑢) ∧ 𝑚(𝑣)for all (𝑢, 𝑣) ∈ 𝑟∗ . 209 Group Decision Making in Conditions of Uncertainty using Fermat’s Weak Fuzzy Graphs and Beal’s Weak Fuzzy Graphs Definition 2.4: A Fermat’s Fuzzy Set (FFS) on a universal set A is a set of 3 tuples of the form F={(u, 𝐼𝐹 (𝑢), 𝑂𝐹(𝑢))} where 𝐼𝐹 (𝑢) and 𝑂𝐹(𝑢) represents the membership and non membership degrees of u A and 𝐼𝐹 (𝑢), 𝑂𝐹(𝑢) satisfy 0 ≤ 𝐼𝐹 𝑛 (𝑢) + 𝑂𝐹 𝑛 (𝑢) ≤ 1 for all u A , n N={1,2,3,..} . Definition 2.5: A Fermat’s fuzzy relation (FFR) R on A A is a set of 3 tuples of the form R = { ( uv, 𝐼𝑅 (𝑢𝑣), 𝑂𝑅 (𝑢𝑣) } where 𝐼𝑅 (𝑢𝑣), and 𝑂𝑅 (𝑢𝑣) represents the membership degree and non membership degree of uv in R and ( ), ( ) R R I uv O uv satisfy 0 ( ) ( ) 1 n n R R I uv O uv +  for all uv A A . FFR need not be symmetric. Hence 𝐼𝑅 (𝑢𝑣) need not be equal to ( ) R I vu . Definition 2.6: A Fermat’s fuzzy graph (FFG(n)) on a non empty set A is a pair G : (𝜎, µ) with 𝜎 as FFS on A and µ as FFR on A such that 𝐼𝜇 (𝑢𝑣) ≤ 𝐼𝜎 (𝑢) ∧ 𝐼𝜎 (𝑣), 𝑂𝜇 (𝑢𝑣) ≥ 𝑂𝜎 (𝑢) ∨ 𝑂𝜎 (𝑣) and 0 ( ) ( ) 1 n n I uv O uv    +  for all u,v A , n N={1,2,3,..} .where 𝐼𝜇 : A A →[0,1] and 𝑂𝜇 : A A →[0,1] represents the membership and non membership functions of µ respectively. Definition 2.7: A Fermat’s fuzzy preference relation (FFPR) on the set of nodes N ={x1, x2, … xn} is represented by a matrix M = (mij)nxn, where mij = ( xixj, I(xixj) , O(xixj) ) for all i,j =1,2,3..n. Let mij = (Iij ,Oij) where Iij indicates the degree to which the node xi is preferred to node xj and Oij denotes the degree to which the node xi is not preferred to the node xj and 𝜋𝑖𝑗 = √1 − 𝐼𝑖𝑗 𝑛 − 𝑂𝑖𝑗 𝑛𝑛 is interpreted as hesitancy degree ,with the conditions, Iij ,Oij [0,1], 0 ≤ 𝐼𝑖𝑗 𝑛 + 𝑂𝑖𝑗 𝑛 ≤ 1 , Iij = Oji , Iii = Oii = 0.5 for all i,j =1,2,3..n. Definition 2.8: A Fermat’s fuzzy graph G : (𝜎, µ) is said to be Fermat’s Strong fuzzy Graph FSFG(n) with underlying crisp graph G*: (𝜎∗, µ∗) 𝑖𝑓 𝐼𝜇 (𝑢𝑣) = 𝐼𝜎 (𝑢) ∧ 𝐼𝜎 (𝑣), 𝑂𝜇 (𝑢𝑣) = 𝑂𝜎 (𝑢) ∨ 𝑂𝜎 (𝑣) for all uv µ ∗ Definition 2.9: A Fermat’s fuzzy graph G : (𝜎, µ) is said to be Fermat’s Weak Fuzzy Graph FWFG(n) with underlying crisp graph G*: (𝜎∗, µ∗) 𝑖𝑓 𝐼𝜇 (𝑢𝑣) < 𝐼𝜎 (𝑢) ∧ 𝐼𝜎 (𝑣), 𝑂𝜇 (𝑢𝑣) > 𝑂𝜎 (𝑢) ∨ 𝑂𝜎 (𝑣) for all uv µ ∗ Definition 2.10 : A Fermat’s fuzzy graph G : (𝜎, µ) is said to be complete FFG with underlying crisp graph G*: (𝜎∗, µ∗) 𝑖𝑓 𝐼𝜇 (𝑢𝑣) = 𝐼𝜎 (𝑢) ∧ 𝐼𝜎 (𝑣), 𝑂𝜇 (𝑢𝑣) = 𝑂𝜎 (𝑢) ∨ 210 T.M.Nishad, B.Mohamed Harif and A Prasanna 𝑂𝜎 (𝑣) for all u,v 𝜎 ∗. 3.Modeling of Group Decision Making Problem Example 3.1: Mr. X from India wish to invest money in any of the following 5 schemes that helps him better financial security in future. 1. Public Provident Fund S1 2. National Saving Certificate S2 3. Atal Pension Yojana S3 4. National Pension Scheme S4 5. Sovereign Gold Bonds S5 He consulted with 4 experts and they advised the merits and demerits of each particular scheme comparing with other. The aggregate of information FFPR is prepared as relation matrices.How can he select the best Scheme? Modeling: Suppose the 5 schemes are S1,S2,S3,S4 and S5. Consider the discrete set of alternatives A = {S1,S2,S3,S4,S5}.Since the alternatives are present , assign the membership degree as 1 and non membership degree 0 to each alternatives. Consider the set of experts as {E1,E2,E3,E4}. Since each experts gives the acceptance and rejection reasons comparing every pair of alternatives, the aggregate of information FFPR can be represented as relation matrices. This data represents a FFG(n). If in the given FFPRs , all the membership values are in (0,1) and non membership values are greater than 0 then the given FFG(n) will be FWFG(n). 4. Decision tools for Fermat’s Fuzzy Graph In decision making, the Optimal Score having maximum rank is considered as best choice. The scores to rank the alternatives can be calculated using score function . Here is the collective Fermat’s Fuzzy Element which can be obtained using Fermat’s Fuzzy Weighted Averaging Operator FFWA. The weight of each expert can be obtained using deviations of each experts and the deviations can be calculated from difference matrices. The entries in difference matrix is calculated using Fermats Fuzzy Hamming distance between Fermat’s Fuzzy Elements. 4.1 Fuzzy Averaging operator FFA Fermat’s Fuzzy Element (FFE) indicates preference of each expert Ek over each pair of alternatives. It is determined using Fermat’s Fuzzy Averaging operator FFA ( )iS p ip ( ) ( ) ( )( ) ( ) ( ) 1 1 1 2 1 1 , ,...., 1 1 , , 1, 2, 3,.., . ij ij m mm m k k k n n i i im j j FFA p p p I O i m   = =        = − − =              ( )k i p 211 Group Decision Making in Conditions of Uncertainty using Fermat’s Weak Fuzzy Graphs and Beal’s Weak Fuzzy Graphs FFE is used in calculation of FFWA. 4.2 Fermat’s Fuzzy Hamming distance between FFEs. From the given FFPR, where are hesitancy degree. 4.3 Difference matrix 4.4 Average Values of Difference Matrix The equation to determine average values of difference matrix 4.5 Deviation of expert Er from remaining experts The equation to determine deviation of expert Er from remaining experts 4.6 Weight of experts wr. The equation to determine weight of experts wr. 4.7 Fermat’s Fuzzy Weighed Averaging operator FFWA Fermat’s Fuzzy Weighed Averaging operator FFWA to compute collective Fermat’s Fuzzy Element pi over other alternatives is 4.8 Score Function Score function to rank the alternatives , 1, 2,..., i u i m= , ( ) ( )( )( ) ( ) , , 1, 2,..., . i i n n i p p i i i S p I O I u O u u i m   = − − = ( ) ( ) ( ) ( )( )1 2, ,...., , 1, 2, 3,.., . k k k k i i i im p FFA p p p i m= = ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 , , 2 l k l k l k ij ij p p p pij ij ij ij l k n n n n n n ij ij p p d p p I I O O         = − + − + −      ( ) ( ) ( ) ( ) ( ) ( ) 1 , 1l k l l k kij ij p p p p ij ij ij ij n n n n n n p p I O I O      = − − = − − ( )( ) ( ) ( )( ),lk l klk ij ij ij mxm mxm D d d p p= = ( ) 2 1 1 1 m m lk lk ij i j d d m = = =  1, s r rk k k r d d =  =  ( ) ( ) 1 1 1 , 1, 2,.., . r r s r r d w r s d − − = = =  ( ) ( ) ( )( ) ( ) ( ) ( )1 2 1 1 , ,...., 1 1 , , k k i ik k p p i i w w s s s n n i i i i p p k k p FFWA p p p I O I O   = =        = = − − =             ( )iS p 212 T.M.Nishad, B.Mohamed Harif and A Prasanna 5. Illustration of a Practical Example In example 3.1, Suppose the aggregate of information FFPR is given as following relation matrices. Data 1. The information from E1 in the form of Relation Matrix. Data 2. The information from E2 in the form of Relation Matrix. Data 3. The information from E3 in the form of Relation Matrix. Data 4. The information from E4 in the form of Relation Matrix. The above data represents a FFG(n). Among the relations,0.8+0.8= 0.9+0.7= 0.7+0.9 = 1.6 is the maximum sum among measures of acceptance and corresponding rejection. Since there is more than one pair with the same sum, we break the tie by comparing the sum of powers and selecting the pair that brings maximum sum.Here 0.82+0.82 =1.28 < ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 5 0.5, 0.5 0.6, 0.8 0.7, 0.9 0.6, 0.9 0.3, 0.91 0.8, 0.6 0.5, 0.5 0.5, 0.8 0.6, 0.7 0.7, 0.92 0.9, 0.7 0.8, 0.5 0.5, 0.5 0.4, 0.9 0.8, 0.43 0.9, 0.6 0.7, 0.6 0.9, 0.4 0.5, 0.5 0.8, 0.54 5 0.9, 0.3 0.9, 0.7 0.4, 0.8 0.5, 0.8 0.5, 0.5 C C C C C C C C C C                  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 5 0.5, 0.5 0.6, 0.7 0.7, 0.8 0.8, 0.8 0.3, 0.81 0.7, 0.6 0.5, 0.5 0.5, 0.8 0.6, 0.7 0.7, 0.82 0.8, 0.7 0.8, 0.5 0.5, 0.5 0.4, 0.9 0.8, 0.53 0.8, 0.8 0.7, 0.6 0.9, 0.4 0.5, 0.5 0.8, 0.54 5 0.8, 0.3 0.8, 0.7 0.5, 0.8 0.5, 0.8 0.5, 0.5 C C C C C C C C C C                  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 5 0.5, 0.5 0.5, 0.8 0.6, 0.9 0.7, 0.9 0.4, 0.91 0.8, 0.5 0.5, 0.5 0.5, 0.8 0.6, 0.7 0.5, 0.92 0.9, 0.6 0.8, 0.5 0.5, 0.5 0.4, 0.9 0.6, 0.43 0.9, 0.7 0.7, 0.6 0.9, 0.4 0.5, 0.5 0.8, 0.54 5 0.9, 0.4 0.9, 0.5 0.4, 0.6 0.5, 0.8 0.5, 0.5 C C C C C C C C C C                  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 5 0.5, 0.5 0.6, 0.8 0.7, 0.9 0.7, 0.9 0.3, 0.91 0.8, 0.6 0.5, 0.5 0.5, 0.8 0.7, 0.7 0.7, 0.92 0.9, 0.7 0.8, 0.5 0.5, 0.5 0.4, 0.8 0.8, 0.43 0.9, 0.7 0.7, 0.7 0.8, 0.4 0.5, 0.5 0.8, 0.54 5 0.9, 0.3 0.9, 0.7 0.4, 0.8 0.5, 0.8 0.5, 0.5 C C C C C C C C C C                  213 Group Decision Making in Conditions of Uncertainty using Fermat’s Weak Fuzzy Graphs and Beal’s Weak Fuzzy Graphs 0.92+0.72 =1.3. So we consider sum of higher powers of 0.9 and 0.7 till we get a sum ≤1. Note that 0.93+0.73 =1.072 > 1.But 0.94+0.74 = 0.8962 < 1 .So the FFG(n) is FFG(4).Since alternatives are present , the membership value 1 and non membership value 0 have to be assigned to each alternatives. In the given FFRs , all the membership and non membership values are in (0,1). Hence the given FFG(4) is FWFG(4). Fermat’s Fuzzy Eelements are P1 (1) = (0.5854,0.7816) , P2 (1) = (0.6624,0.6853) , P3 (1) = (0.7732,0.5753) , P4 (1) = (0.8196,0.5144) , P5 (1) = (0.7786,0.5827) P1 (2) = (0.6544,0.7090) , P2 (2) = (0.6231,0.6694) , P3 (2) = (0.7301,0.6015) , P4 (2) = (0.7896,0.5448) , P5 (2) = (0.6855,0.5827) P1 (3) = (0.5725,0.7816) , P2 (3) = (0.6304,0.6608) , P3 (3) = (0.7435,0.5578) , P4 (3) = (0.8196,0.5305) , P5 (3) = (0.7786,0.5448) P1 (4) = (0.6129,0.7816) , P2 (4) = (0.6803,0.6853) , P3 (4) = (0.7732,0.5619) , P4 (4) = (0.7896,0.5471) , P5 (4) = (0.7786,0.5827) From the difference matrices and the average values of difference matrices we get the deviations d1= 0.196528, d2= 0.341216, d3= 0.261024 and d4= 0.242128. Then the weights of experts are w1= 0.31842, w2= 0.18340, w3= 0.23974 and w4= 0.25845. Now the collective Fermat’s Fuzzy Elements are p1= ( Ip1, Op1) = (0.60466,0.76775), p2 = ( Ip2, Op2) = (0.65365,0.67642), p3 = ( Ip3, Op3) = (0.75922,0.57224), p4 = ( Ip4, Op4) = (0.80715,0.53211) and p5 = ( Ip5, Op5) = (0.76520,0.57338). The corresponding score function gives the following scores S(p1) = - 0.21377, S(p2) = - 0.02680, S(p3) = 0.22503, S(p4) = 0.34427 and S(p5) = 0.23476 Since S(p4) is the maximum score ,the best Choice is S4, the National Pension Scheme. 6. Beal’s Fuzzy Graph BFG(m,n) If the membership value of acceptance (or rejection) is given a limit ( say α ) then the membership value ( say β) of rejection (or acceptance ) is assumed to be governed by the in equation βn ≤ 1- αm for some m,n N={1,2,3,..} .Therefore the generalization of FFG(n) has importance. 214 T.M.Nishad, B.Mohamed Harif and A Prasanna Definition 6.1: A Beal’s Fuzzy Set (BFS) on a universal set A is a set of 3 tuples of the form F={(u, 𝐼𝐹 (𝑢), 𝑂𝐹(𝑢))} where 𝐼𝐹 (𝑢) and 𝑂𝐹 (𝑢) represents the membership and non membership degrees of u A and 𝐼𝐹 (𝑢), 𝑂𝐹(𝑢) satisfy 0 ≤ 𝐼𝐹 𝑚(𝑢) + 𝑂𝐹 𝑛 (𝑢) ≤ 1 for all u A , m,n N={1,2,3,..} . Definition 6.2: A Beal’s fuzzy relation (BFR) R on A A is a set of 3 tuples of the form R = { ( uv, 𝐼𝑅 (𝑢𝑣), 𝑂𝑅 (𝑢𝑣) } where 𝐼𝑅 (𝑢𝑣), and 𝑂𝑅 (𝑢𝑣) represents the membership degree and non membership degree of uv in R and ( ), ( ) R R I uv O uv satisfy 0 ≤ 𝐼𝑅 𝑚(𝑢𝑣) +)𝑂𝑅 𝑛(𝑢𝑣) ≤ 1 for all uv A A . BFR need not be symmetric. Hence 𝐼𝑅 (𝑢𝑣) need not be equal to ( )RI vu . Definition 6.3:A Beal’s fuzzy graph BFG(m,n) on a non empty set A is a pair G : (𝜎, µ) with 𝜎 as BFS on A and µ as BFR on A such that 𝐼𝜇 (𝑢𝑣) ≤ 𝐼𝜎 (𝑢) ∧ 𝐼𝜎 (𝑣), 𝑂𝜇 (𝑢𝑣) ≥ 𝑂𝜎 (𝑢) ∨ 𝑂𝜎 (𝑣) and 0 ≤ 𝐼µ 𝑚(𝑢𝑣) +)𝑂µ 𝑛(𝑢𝑣) ≤ 1 for all u,v A , m,n N={1,2,3,..} .where 𝐼𝜇 : A A →[0,1] and 𝑂𝜇 : A A →[0,1] represents the membership and non membership functions of µ respectively. Definition 6.4: A Beal’s fuzzy graph G : (𝜎, µ) is said to be Beal’s Strong fuzzy Graph BSFG(m,n) with underlying crisp graph G*: (𝜎∗, µ∗) 𝑖𝑓 𝐼𝜇 (𝑢𝑣) = 𝐼𝜎 (𝑢) ∧ 𝐼𝜎 (𝑣), 𝑂𝜇 (𝑢𝑣) = 𝑂𝜎 (𝑢) ∨ 𝑂𝜎 (𝑣) for all uv µ ∗ . Definition 6.5: A Beal’s fuzzy graph G : (𝜎, µ) is said to be Beal’s Weak Fuzzy Graph BWFG(m,n) with underlying crisp graph G*: (𝜎∗, µ∗) 𝑖𝑓 𝐼𝜇 (𝑢𝑣) < 𝐼𝜎 (𝑢) ∧ 𝐼𝜎 (𝑣), 𝑂𝜇 (𝑢𝑣) > 𝑂𝜎 (𝑢) ∨ 𝑂𝜎 (𝑣) for all uv µ ∗ Definition 6.6 : A Beal’s fuzzy graph G : (𝜎, µ) is said to be complete BFG with underlying crisp graph G*: (𝜎∗, µ∗) 𝑖𝑓 𝐼𝜇 (𝑢𝑣) = 𝐼𝜎 (𝑢) ∧ 𝐼𝜎 (𝑣), 𝑂𝜇 (𝑢𝑣) = 𝑂𝜎 (𝑢) ∨ 𝑂𝜎 (𝑣) for all u,v 𝜎 ∗. Theorem 6.1: When m = n, BFG (m,n) FFG(n) and BFG (1,1) FFG(1) which is an intuitionistic fuzzy graph. Proof. Directly follows from the definitions. i.e, Beal’s Fuzzy graph is generalization of Fermat’s Fuzzy Graph and Fermat’s Fuzzy Graph is generalization of Intuitionistic Fuzzy Graph. Theorem 6.2: BWFG (m-1,n-1)  BWFG(m,n) but the converse is not true. Proof. Let G : (𝜎, µ) be a BWFG (n-1) with 𝜎 as BFS on A and µ as BFR on A . Since 𝐼𝜇 (𝑢𝑣) < 1 , 𝐼µ 𝑚−1(𝑢𝑣) < 1  𝐼µ 𝑚(𝑢𝑣) < 𝐼µ 𝑚−1(𝑢𝑣) < 1, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑚 𝑁 → (1) Similarly since 1 1 ( ) 1, ( ) 1 ( ) ( ) n n n O uv O uv O uv O uv     − −       215 Group Decision Making in Conditions of Uncertainty using Fermat’s Weak Fuzzy Graphs and Beal’s Weak Fuzzy Graphs Therefore 𝐼µ 𝑚−1(𝑢𝑣) +)𝑂µ 𝑛−1(𝑢𝑣) ≤ 1  𝐼µ 𝑚(𝑢𝑣) + 𝑂µ 𝑛(𝑢𝑣) ≤ 1. Hence BWFG(m-1,n-1)  BWFG(m,n). It is obvious from equation (1) that the converse is not true. 7 Conclusion In this article some fundamental concepts of Fuzzy Graphs and Fermat’s Fuzzy Graphs are reviewed.The decision tools are generalized for Fermat’s Fuzzy Graphs. Application of Fermats Weak Fuzzy Graph in modeling group decision making problem is illustrated with a practical example. The fundamental concepts of Beal’s Fuzzy Graph are developed. The applications of FFG(n) and BFG(m,n) in various fields of science, social science and engineering are under research. 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