Decision Making: Applications in Management and Engineering Vol. 5, Issue 2, 2022, pp. 176-200. ISSN: 2560-6018 eISSN: 2620-0104 DOI: https://doi.org/10.31181/dmame0311072022b * Corresponding author. broumisaid78@gmail.com, s.broumi@flbenmsik.ma E-mail addresses: sundareswaranr@ssn.edu.in (R. Sundareswaran), shanmugapriyma@ssn.edu.in (M. Shanmugapriya), giorgio.nordo@unime.it (G. Nordo), taleamohamed@yahoo.fr (M. Talea), assiabakali@yahoo.fr (A. Bakali), fsmarandache@gmail.com (F. Smarandache) INTERVAL- VALUED FERMATEAN NEUTROSOPHIC GRAPHS Said Broumi1*, Raman Sundareswaran2, Marayanagaraj Shanmugapriya3, Giorgio Nordo4, Mohamed Talea1, Assia Bakali5, and Florentin Smarandache6 1 Laboratory of Information Processing, Faculty of Science Ben M’Sik, University of Hassan II, Casablanca, Morocco & Regional Center for the Professions of Education and Training(C.R.M.E.F), Casablanca,Morocco. 2,3Department of Mathematics, Sri Sivasubramaniya Nadar College of Engineering, India 4Dipartimento di scienze Matematiche e Informatiche, scienze Fisiche e scienze della Terra dell'Università degli Studi di Messina Viale Ferdinando Stagno d'Alcontres, Italy 5 Ecole Royale Navale-Boulevard Sour Jdid, Morocco 6Department of Mathematics, University of New Mexico, USA Received: 2 May 2022; Accepted: 29 June 2022; Available online: 12 July 2022. Original scientific paper Abstract: In this work, we define Interval-valued Fermatean neutrosophic graphs (IVFNS) and present some operations on Interval-valued Fermatean neutrosophic graphs. Further, we introduce the concepts of Regular interval- valued Fermatean neutrosophic graphs, Strong interval-valued Fermatean neutrosophic graphs, Cartesian, Composition, Lexicographic product of interval-valued Fermatean neutrosophic graphs. Finally, we give the applications of Interval-valued Fermatean neutrosophic graphs. Key words: Interval-valued Fermatean Fuzzy sets, Interval-valued Fermatean Neutrosophic sets, Interval-valued Fermatean Neutrosophic graphs mailto:s.broumi@flbenmsik.ma mailto:shanmugapriyma@ssn.edu.in Interval- valued Fermatean Neutrosophic Graphs 177 1. Introduction The concept of neutrosophic set theory was proposed by Jun (2017). The idea of neutrosophic set which is a generalization of the fuzzy set (Zadeh, 1965), intuitionistic fuzzy set (Atanassov, 1986). The neutrosophic sets are characterized by a truth function (T), an indeterminate function (I) and a false function (F) independently. Smarandache (2019) introduced the concept of spherical neutrosophic oversets as generalization of spherical fuzzy sets. By bending the concept of single valued neutrosophic set and graph theory, different classes of neutrosophic graphs is discussed by Broumi (2016) and many works available in the literature (Broumi et al., 2016a, 2016b, 2016c, 2016d, 2022). Nagarajan et al. (2019) investigated the interval- valued neutrosophic graphs and its applications. Recently, Ajay et al. (2020, 2021) extended the concept of Pythagorean neutrosophic sets to graphs and called it Pythagorean neutrosophic graph (PNG) and investigated some of their properties. The same authors presented the idea of labelling of Pythagorean neutrosophic fuzzy graphs and investigate their properties. Ajay et al. (2022) studied the concept of regularity in PNG and introduced the ideas of regular, full edge regular, edge regular, and partially edge regular Pythagorean Neutrosophic graphs. In addition, a new MCDM method has been introduced using the Pythagorean neutrosophic graphs with an illustrative example. By integrating the concepts pythagorean neutrosophic fuzzy graph and Dombi operator. Furthermore, Ajay et al. (2021) proposed a new extension of neutrosophic graph called Pythagorean Neutrosophic Dombi fuzzy graphs (PNDFG) and suggested some basic operations of PNDFG and computed the degree and total degree of a vertex of PNDFG. Akalyadevi et al. (2022) introduced the concept of spherical neutrosophic graph coloring and discussed some of their important properties also they suggested the chromatic number of spherical neutrosophic graph as a crisp number. Duleba et al. (2021) applied the concept of Interval-valued spherical fuzzy AHP method to public transportation problem. Aydın et al. (2021) proposed a novel fuzzy MULTIMOORA method based on interval-valued spherical fuzzy sets to evaluate companies that are using Industry 4.0 technologies. Lathamaheswari et al. (2021) proposed the concept of Interval Valued Spherical Fuzzy Aggregation Operators and applied it for solving a Decision-Making Problem. Kutlu Gündoğdu et al. (2021) extended spherical fuzzy analytic hierarchy process to interval-valued spherical fuzzy AHP (IVSF-AHP) method and applied it to compare the service performances of several hospitals. Kutlu Gündoğdu et al. (2019) presented the idea of Spherical fuzzy sets (SFS) as an integration of Pythagorean fuzzy sets and neutrosophic sets. Smarandache (2017) proposed the concept of Spherical Neutrosophic Numbers. Senapati et al. (2019) defined basic operators over the FFSs. On the other hand, division, and subtraction operations on FFSs were proposed. Donghai Liu et al. (2019) focused on a distance measure for Fermatean fuzzy linguistic term sets. Ganie et al. (2022) proposed some novel distance measures for Fermatean fuzzy sets using t-conorms. On the other hand, Jeevaraj et al. (2021) proposed the concept of interval-valued Fermatean fuzzy sets (IVFFSs) and establishes some Mathematical operations on the class of IVFFSs. A new total ordering principle on the class of IVFFNs is introduced. They implemented the interval-valued Fermatean fuzzy TOPSIS (IVFFTOPSIS) method for solving multi-criteria decision-making problems. Based on neutrosophic Pythagorean sets, Stephen et al. (2021) introduced the concept of interval-valued neutrosophic Pythagorean sets with dependent interval valued Pythagorean components and discussed some of its properties. Recently, Lakhwani et al. (2022) introduced a novel concept of Dombi neutrosophic graph and presented some kinds of Dombi neutrosophic graph such as a regular Dombi neutrosophic graph, Said Broumi et al./Decis. Mak. Appl. Manag. Eng. 5 (2) (2022) 176-200 178 strong Dombi neutrosophic graph, complete Dombi neutrosophic graph, and complement Dombi neutrosophic graph and described some of their properties, also, and discussed some operations on Dombi neutrosophic graphs are defined. In this paper, we present the concept of Interval-valued Fermatean neutrosophic graphs (IVFNG) and the concepts of Regular interval-valued Fermatean neutrosophic graphs, Strong interval-valued Fermatean neutrosophic graphs, Cartesian, Composition, Lexicographic product of interval-valued Fermatean neutrosophic graphs. We also introduce some theorems and examples on IVFNG’s Finally, we give the applications of Interval-valued Fermatean neutrosophic graphs. 2. Preliminaries The extension of crisp set theory with membership degree is known as Fuzzy set theory in which each element of the set gets a real number between 0 and 1. But in many real time situations, it is not always possible to give an exact degree of membership because of lack of knowledge, vague information, and so forth. To overcome this problem, we can use interval-valued fuzzy sets, which assign to each element a closed interval which approximates the “real,” but unknown, membership degree. The length of this interval is a measure for the uncertainty about the membership degree. An interval number I is an interval [𝑐 −, 𝑐+] with 0 ≤ 𝑐 − ≤ 𝑐+ ≤ 1. The interval [c, 𝑐] is identified with the number 𝑐 ∈ [0, 1]. Let 𝐼[0, 1] be the set of all closed subintervals of [0, 1]. An extension of fuzzy sets by Zadeh (1965), Interval- valued fuzzy sets which stated that the values of the membership degrees are intervals of numbers instead of the numbers. It provides a more sufficient information about uncertainty than traditional fuzzy sets. In this section, we provide all the basic definitions of interval valued sets and its corresponding graphs. Table 1 depicts the types of sets and graphs for interval-valued fuzzy and neutrosophic environments. Table 1. Different types of Interval-valued sets and its graphs Type of Sets Definition Type of Graphs Definition Interval-valued Fuzzy set (IVFS) -Zadeh, 1975 𝐴 = {(𝑥, [𝜇𝐴 −(𝑥), 𝜇𝐴 +(𝑥)]): 𝑥 ∈ 𝑉} Interval- valued Fuzzy graph (IVFG) - Muhammad Akram, Wieslaw A. Dudek, 2011. G = (A, B), where A = [𝜇𝐴 −(𝑥), 𝜇𝐴 +(𝑥)] is an interval-valued fuzzy set on V and B = [𝜇𝐵 −(𝑥), 𝜇𝐵 +(𝑥)]is an interval-valued fuzzy relation on E. Interval-valued Intuitionistic Fuzzy set (IVIFS) - Atanassov, K., Gargov, G., 1989 𝐴 = {(𝑥, [𝑇𝐴 −(𝑥), 𝑇𝐴 +(𝑥)]): ∈ 𝑉}; 𝐵 = {(𝑥, [𝐹𝐴 −(𝑥), 𝐹𝐴 +(𝑥)]): 𝑥 ∈ 𝑉} such that 0 ≤ 𝑇𝐴 +(𝑥) + 𝐹𝐴 +(𝑥) ≤ 1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥𝜖 𝑋 Interval- valued Intuitionistic Fuzzy graph (IVIFG) - S. N. Mishra and A. Pal, 2013  𝜇𝐴: 𝑉 → 𝐷[0,1]; 𝜂𝐴: 𝑉 → 𝐷[0,1] such that 0 ≤ 𝜇𝐴(𝑥) + 𝜂𝐴(𝑥) ≤ 1 , ∀ 𝑥 ∈ 𝑉  𝜇𝐵 : 𝐸 ⊆ 𝑉 × 𝑉 → 𝐷[0,1] ; 𝜂𝐵 : 𝐸 ⊆ 𝑉 × 𝑉 → 𝐷[0,1] 𝜇𝐵 −((𝑥, 𝑦)) ≤ min( 𝜇𝐴 −(𝑥), 𝜇𝐴 −(𝑦)); 𝜂𝐵 −((𝑥, 𝑦)) ≥ min( 𝜂𝐴 −(𝑥), 𝜂𝐴 −(𝑦)) 𝜇𝐵 +((𝑥, 𝑦)) ≤ min( µ𝐴 +(𝑥), µ𝐴 +(𝑦)); 𝜂𝐵 +((𝑥, 𝑦)) ≥ min( 𝜂𝐴 +(𝑥), 𝜂𝐴 +(𝑦)) such that 0 ≤ 𝜇𝐵 +((𝑥, 𝑦)) + 𝜂𝐵 +((𝑥, 𝑦)) ≤ 1 , ∀(𝑥, 𝑦) ∈ 𝐸 Interval-valued Neutrosophic set (IVNS) - Said Broumi , Mohamed Talea , Assia Bakali , Florentin For each point 𝑥 ∈ 𝑋, we have that 𝑇𝐴(𝑥) = [𝑇𝐴 −(𝑥), 𝑇𝐴 +(𝑥)], 𝐼𝐴(𝑥) = [𝐼𝐴 −(𝑥), 𝐼𝐴 +(𝑥)], 𝐹𝐴(𝑥) = [𝐹𝐴 −(𝑥), 𝐹𝐴 +(𝑥)] ⊆ [0, 1] and 0 ≤ 𝑇𝐴(𝑥) + 𝐼𝐴(𝑥) + 𝐹𝐴(𝑥) ≤ 3. Interval- valued Neutrosophic graph (IVNG) - Said Broumi, Mohamed Talea, Assia Bakali, 𝐺 = (𝐴, 𝐵), where 𝐴 =< [𝑇𝐴 −, 𝑇𝐴 +], [𝐼𝐴 −, 𝐼𝐴 +], [𝐹𝐴 −, 𝐹𝐴 +] > is an interval- valued neutrosophic set on V; and 𝐵 = < [𝑇𝐵 −, 𝑇𝐵 +], [𝐼𝐵 −, 𝐼𝐵 +], [𝐹𝐵 −, 𝐹𝐵 +] > 𝑇𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝑇𝐵 +: 𝑉 × 𝑉 → [0, 1], 𝐼𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝐼𝐵 +: 𝑉 × 𝑉 → [0, 1] and 𝐹𝐵 −: 𝑉 × 𝑉 → [0,1], 𝐹𝐵 +: 𝑉 × 𝑉 → [0, 1] are such that 𝑇𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 −(𝑣𝑖 ), 𝑇𝐴 −(𝑣𝑗 )], Interval- valued Fermatean Neutrosophic Graphs 179 Definition 2.1 (Akram et al., 2013) The Interval-valued Fuzzy Set (IVFS) 𝐴 in 𝑉 is defined by = {(𝑥, { µ𝐴 − (𝑥), µ𝐴 + (𝑥))} ∶ 𝑥 ∈ 𝑉 } , where µ𝐴 − (𝑥) and µ𝐴 + (𝑥) are fuzzy subsets of 𝑉 such that µ𝐴 − (𝑥) ≤ µA + (x) for all 𝑥 ∈ 𝑉. For any two interval-valued sets 𝐴 = [µ𝐴 − (𝑥), µ𝐴 + (𝑥)] and 𝐵 = [µ𝐵 − (𝑥), µ𝐵 + (𝑥)] in V. Define: • 𝐴 ⋃ 𝐵 = {(𝑥, 𝑚𝑎𝑥(µ𝐴 − (𝑥), µ𝐵 − (𝑥)), 𝑚𝑎𝑥(µ𝐴 + (𝑥), µ𝐵 + (𝑥))) ∶ 𝑥 ∈ 𝑉}, • 𝐴 ⋂ 𝐵 = {(𝑥, 𝑚𝑖𝑛(µ𝐴 − (𝑥), µ𝐵 − (𝑥)), 𝑚𝑖𝑛(µ𝐴 + (𝑥), µ𝐵 + (𝑥))) ∶ 𝑥 ∈ 𝑉}. Smarandache, (2016) Florentin Smarandache, 2016 𝑇𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 +(𝑣𝑖 ), 𝑇𝐴 +(𝑣𝑗 )], 𝐼𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐼𝐵 − (𝑣𝑖 ), 𝐼𝐵 −(𝑣𝑗 )], 𝐼𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐼𝐵 +(𝑣𝑖 ), 𝐼𝐵 +(𝑣𝑗 )], 𝐹𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐹𝐵 −(𝑣𝑖), 𝐹𝐵 −(𝑣𝑗 )], 𝐹𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ 𝑚𝑎𝑥[𝐹𝐵 +(𝑣𝑖 ), 𝐹𝐵 +(𝑣𝑗 )] Interval-valued Pythagorean Fuzzy set (IVPFS) - F. Teng, Z. Liu, and P. Liu, (2018). For each point 𝑥 ∈ 𝑋, we have that 𝑇𝐴(𝑥) = [𝑇𝐴 −(𝑥), 𝑇𝐴 +(𝑥)], 𝐹𝐴(𝑥) = [𝐹𝐴 −(𝑥), 𝐹𝐴 +(𝑥)] ⊆ [0, 1] and 0 ≤ 𝑇𝐴 +(𝑥)2 + 𝐹𝐴 +(𝑥)2 ≤ 1. Interval- valued Pythagorean Fuzzy graph (IVPFG) - Mohamed S.Y., Ali A.M.,2018 𝐺 = (𝐴, 𝐵), where 𝐴 =< [𝑇𝐴 −, 𝑇𝐴 +], [𝐹𝐴 −, 𝐹𝐴 +] > is an interval-valued neutrosophic set on V; and 𝐵 = < [𝑇𝐵 −, 𝑇𝐵 +], [𝐹𝐵 −, 𝐹𝐵 +] > 𝑇𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝑇𝐵 +: 𝑉 × 𝑉 → [0, 1], and 𝐹𝐵 −: 𝑉 × 𝑉 → [0,1], 𝐹𝐵 +: 𝑉 × 𝑉 → [0, 1] are such that 𝑇𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 −(𝑣𝑖 ), 𝑇𝐴 −(𝑣𝑗 )], 𝑇𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 +(𝑣𝑖 ), 𝑇𝐴 +(𝑣𝑗 )], 𝐹𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐹𝐵 −(𝑣𝑖), 𝐹𝐵 −(𝑣𝑗 )], 𝐹𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐹𝐵 +(𝑣𝑖), 𝐹𝐵 +(𝑣𝑗 )] such that 0 ≤ 𝑇𝐴 +(𝑥)2 + 𝐹𝐴 +(𝑥)2 ≤ 1. Interval-valued Fermatean Fuzzy set (IVFFS) - Jeevaraj S, (2021) For each point 𝑥 ∈ 𝑋, we have that 𝑇𝐴(𝑥) = [𝑇𝐴 −(𝑥), 𝑇𝐴 +(𝑥)], 𝐹𝐴(𝑥) = [𝐹𝐴 −(𝑥), 𝐹𝐴 +(𝑥)] ⊆ [0, 1] and 0 ≤ 𝑇𝐴 +(𝑥)3 + 𝐹𝐴 +(𝑥)3 ≤ 1. Interval- valued Fermatean Fuzzy graph (IVFFG) 𝐺 = (𝐴, 𝐵), where 𝐴 =< [𝑇𝐴 −, 𝑇𝐴 +], [𝐹𝐴 −, 𝐹𝐴 +] > is an interval-valued neutrosophic set on V; and 𝐵 = < [𝑇𝐵 −, 𝑇𝐵 +], [𝐹𝐵 −, 𝐹𝐵 +] > 𝑇𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝑇𝐵 +: 𝑉 × 𝑉 → [0, 1], and 𝐹𝐵 −: 𝑉 × 𝑉 → [0,1], 𝐹𝐵 +: 𝑉 × 𝑉 → [0, 1] are such that 𝑇𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 −(𝑣𝑖 ), 𝑇𝐴 −(𝑣𝑗 )], 𝑇𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 +(𝑣𝑖 ), 𝑇𝐴 +(𝑣𝑗 )], 𝐹𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐹𝐵 −(𝑣𝑖), 𝐹𝐵 −(𝑣𝑗 )], 𝐹𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ 𝑚𝑎𝑥[𝐹𝐵 +(𝑣𝑖 ), 𝐹𝐵 +(𝑣𝑗 )] such that 0 ≤ 𝑇𝐴 +(𝑥)3 + 𝐹𝐴 +(𝑥)3 ≤ 1. Interval-valued Fermatean Neutrosophic set (IVFNS) – Said Broumi, Raman Sundareswaran, Marayanagaraj Shanmugapriya, Giorgio Nordo Mohamed Talea, Assia Bakali, and Florentin Smarandache, (2022) A={〈𝑥, 𝑇𝐴(𝑥), IA(𝑥), FA(𝑥)〉| 𝑥 ∈ X } where 𝑇𝐴(𝑥) = [TA −(𝑥) , 𝑇𝐴 +(𝑥)], IA(𝑥)=[IA −(𝑥) , IA +(x)] and FA(𝑥) = [𝐹𝐴 −(𝑥), FA +(𝑥)], 𝑇𝐴(𝑥): 𝑋 → 𝐷[0,1] IA(𝑥): 𝑋 → 𝐷[0,1], 𝐹𝐴(𝑥): 𝑋 → 𝐷[0,1] and 0 ≤ (𝑇𝐴(𝑥)) 𝟑 +(𝐹𝐴(𝑥)) 𝟑 ≤1 and 0 ≤ (𝐼𝐴 (𝑥)) 𝟑 ≤ 1 0 ≤ (𝑇𝐴(𝑥)) 𝟑 +(𝐹𝐴(𝑥)) 𝟑 +(𝐼𝐴(𝑥)) 𝟑 ≤ 2 means 0 ≤ (𝑇𝐴 +(x))𝟑+(𝐼𝐴 +(𝑥))𝟑+(𝐹𝐴 +(𝑥))𝟑 ≤ 2 ∀ 𝑥 ∈ X Interval- valued Fermatean Neutrosophic graph (IVFNG) - Said Broumi, et al., 2022 𝐺 = (𝐴, 𝐵), where 𝐴 =< [𝑇𝐴 −, 𝑇𝐴 +], [𝐼𝐴 −, 𝐼𝐴 +], [𝐹𝐴 −, 𝐹𝐴 +] > is an interval- valued Fermatean neutrosophic set on V; and 𝐵 = < [𝑇𝐵 −, 𝑇𝐵 +], [𝐼𝐵 −, 𝐼𝐵 +], [𝐹𝐵 −, 𝐹𝐵 +] > 𝐸 satisfying the following condition: 𝑇𝐴 − ∶ 𝑉 → [0, 1], 𝑇𝐴 + ∶ 𝑉 → [0, 1], 𝐼𝐴 − ∶ 𝑉 → [0, 1], 𝐼𝐴 +: 𝑉 → [0, 1]𝑎𝑛𝑑 𝐹𝐴 −: 𝑉 → [0, 1], 𝐹𝐴 +: 𝑉 → [0, 1], and 0 ≤ 𝑇𝐴(𝑣𝑖) + 𝐼𝐴(𝑣𝑖 ) + 𝐹𝐴(𝑣𝑖 ) ≤ 3, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑖 ∈ 𝑉 (𝑖 = 1, 2, … , 𝑛 The functions 𝑇𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝑇𝐵 +: 𝑉 × 𝑉 → [0, 1], 𝐼𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝐼𝐵 +: 𝑉 × 𝑉 → [0, 1] and 𝐹𝐵 −: 𝑉 × 𝑉 → [0,1], 𝐹𝐵 +: 𝑉 × 𝑉 → [0, 1] are such that 𝑇𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 −(𝑣𝑖 ), 𝑇𝐴 −(𝑣𝑗 )], 𝑇𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 +(𝑣𝑖 ), 𝑇𝐴 +(𝑣𝑗 )], 𝐼𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐼𝐵 − (𝑣𝑖 ), 𝐼𝐵 −(𝑣𝑗 )], 𝐼𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐼𝐵 +(𝑣𝑖 ), 𝐼𝐵 +(𝑣𝑗 )], 𝐹𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐹𝐵 −(𝑣𝑖), 𝐹𝐵 −(𝑣𝑗 )], 𝐹𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐹𝐵 +(𝑣𝑖), 𝐹𝐵 +(𝑣𝑗 )], denoting the degree of truth-membership, indeterminacy-membership and falsity- membership of the edge (𝑣𝑖 , 𝑣𝑗 ) ∈ 𝐸 respectively, where 0 ≤ 𝑇𝐵 ({𝑣𝑖 , 𝑣𝑗 }) 3 + 𝐼𝐵({𝑣𝑖 , 𝑣𝑗 }) 3 + 𝐹𝐵 ({𝑣𝑖 , 𝑣𝑗 }) 3 ≤ 2 for all {𝑣𝑖 , 𝑣𝑗 } ∈ 𝐸 (𝑖, 𝑗 = 1, 2, … , 𝑛) means 0 ≤ (𝑇𝐵 +(𝑣𝑖 , 𝑣𝑗 )) 𝟑 +(𝐼𝐵 +(𝑣𝑖 , 𝑣𝑗 )) 𝟑 +(𝐹𝐵 +(𝑣𝑖 , 𝑣𝑗 )) 𝟑 ≤ 2 ∀ 𝑥 ∈ X. Said Broumi et al./Decis. Mak. Appl. Manag. Eng. 5 (2) (2022) 176-200 180 Definition 2.2 (Akram et al., 2013) If 𝐺 ∗ = (𝑉, 𝐸) is a graph, then by an Interval-valued Fuzzy Relation (IVFR) 𝐵 on a set 𝐸 we mean an interval-valued fuzzy set such that µ𝐵 − (𝑥𝑦) ≤ 𝑚𝑖𝑛(µ𝐴 − (𝑥), µ𝐴 − (𝑦)), µ𝐵 + (𝑥𝑦) ≤ 𝑚𝑖𝑛(µ𝐴 +(𝑥), µ𝐴 + (𝑦)) for all 𝑥𝑦 ∈ 𝐸. Definition 2.3 (Akram et al., 2013) By an Interval-valued Fuzzy Graph (IVFG) of a graph 𝐺 ∗ = (𝑉, 𝐸) we mean a pair 𝐺 = (𝐴, 𝐵), where 𝐴 = [µ𝐴 − , µ𝐴 + ] is an interval-valued fuzzy set on 𝑉 and 𝐵 = [µ𝐵 − , µ𝐵 + ] is an interval-valued fuzzy relation on 𝐸. Example 2.4 (Akram et al., 2013) Consider a graph 𝐺 ∗ = (𝑉, 𝐸) such that 𝑉 = {𝑥, 𝑦, 𝑧}, 𝐸 = {𝑥𝑦, 𝑦𝑧, 𝑧𝑥}. Let A be an interval-valued fuzzy set of V and B be an interval-valued fuzzy set of 𝐸 ⊆ 𝑉 × 𝑉 defined by 𝐴 = 〈( 𝑥 0.2 , 𝑦 0.3 , 𝑧 0.4 ) , ( 𝑥 0.4 , 𝑦 0.5 , 𝑧 0.5 ) 〉 , 𝐵 = 〈( 𝑥 0.2 , 𝑦 0.3 , 𝑧 0.4 ) , ( 𝑥𝑦 0.3 , 𝑦𝑧 0.4 , 𝑧𝑥 0.4 ) 〉 Figure 1. Interval-Valued Fuzzy Graph G Akram et al. (2013) introduced certain types of interval-valued fuzzy graphs including balanced interval-valued fuzzy graphs, neighbourly irregular interval- valued fuzzy graphs, neighbourly total irregular interval-valued fuzzy graphs, highly irregular interval-valued fuzzy graphs, and highly total irregular interval-valued fuzzy graph. Hossein et al. (2013) define three new operations on interval-valued fuzzy graphs; namely direct product, semi strong product and strong product. Definition 2.5 (Mishra et al., 2013; Ismayil et al., 2014) An Interval-valued Intuitionistic Fuzzy Set (IVIFS) 𝐴 in 𝑋, is given by 𝐴 = { 〈𝑥, 𝜇𝐴(𝑥), 𝜂𝐴(𝑥)〉/ 𝑥𝜖 𝑋} where 𝜇𝐴: 𝑋 → [0, 1], 𝜂𝐴: 𝑋 → 𝐷[0, 1]. The intervals 𝜇𝐴(𝑥) and 𝜂𝐴(𝑥) denote the degree of membership and the degree of non-membership of the element 𝑥 to the set, where 𝜇𝐴(𝑥) = [ 𝜇𝐴 −(𝑥), 𝜇𝐴 +(𝑥)] and 𝜂𝐴(𝑥) = [𝜂𝐴 −(𝑥), 𝜂𝐴 +(𝑥)] with the condition 0 ≤ 𝜇𝐴 +(𝑥) + 𝜂𝐴 +(𝑥) ≤ 1 for all 𝑥𝜖 𝑋. Definition 2.6 (Mishra et al., 2013; Ismayil et al., 2014) An Interval-valued Intuitionistic Fuzzy Graph (IVIFG) with underlying set V is defined to be a pair 𝐺 = (𝐴, 𝐵) where  the functions 𝜇𝐴: 𝑉 → 𝐷[0,1]; 𝜂𝐴: 𝑉 → 𝐷[0,1] denote the degree of membership and non-membership of the element 𝑥 ∈ 𝑉 respectively, such that 0 ≤ 𝜇𝐴(𝑥) + 𝜂𝐴(𝑥) ≤ 1 , ∀ 𝑥 ∈ 𝑉 Interval- valued Fermatean Neutrosophic Graphs 181  the functions 𝜇𝐵 : 𝐸 ⊆ 𝑉 × 𝑉 → 𝐷[0,1] ; 𝜂𝐵 : 𝐸 ⊆ 𝑉 × 𝑉 → 𝐷[0,1] are defined by 𝜇𝐵 −((𝑥, 𝑦)) ≤ min( 𝜇𝐴 −(𝑥), 𝜇𝐴 −(𝑦)) ; 𝜂𝐵 −((𝑥, 𝑦)) ≥ min( 𝜂𝐴 −(𝑥), 𝜂𝐴 −(𝑦)) 𝜇𝐵 +((𝑥, 𝑦)) ≤ min( µ 𝐴 +(𝑥), µ 𝐴 +(𝑦)) ; 𝜂𝐵 +((𝑥, 𝑦)) ≥ min( 𝜂 𝐴 +(𝑥), 𝜂 𝐴 +(𝑦)) such that 0 ≤ 𝜇𝐵 +((𝑥, 𝑦)) + 𝜂𝐵 +((𝑥, 𝑦)) ≤ 1 , ∀(𝑥, 𝑦) ∈ 𝐸 Example 2.7 𝐺 = (𝐴, 𝐵) defined on a graph 𝐺 ∗ = (𝑉, 𝐸) such that 𝑉 = {𝑥, 𝑦, 𝑧}, 𝐸 = {𝑥𝑦, 𝑦𝑧, 𝑧𝑥}, A is an interval valued intuitionistic fuzzy set of 𝑉and let 𝐵 is an interval- valued intuitionistic fuzzy set of 𝐸 ⊆ 𝑉 𝑋 𝑉. here 𝐴 = {〈𝑥, [0.5,0.7], [0.1,0.3]〉, 〈𝑦, [0.6,0.7], [0.1,0.3]〉 , 〈𝑧, [0.4,0.6], [0.2,0.4]〉 } 𝐵 = {〈𝑥𝑦, [0.3,0.6], [0.2,0.4]〉, 〈𝑦𝑧, [0.3,0.5], [0.3,0.4]〉 , 〈𝑥𝑧, [0.3,0.5], [0.2,0.4]〉} Figure 2. Interval-Valued Intuitionistic Fuzzy Graph G Mishra et al. (2013) introduced product of IVIFG and Ismayil et al. (2014) defined On Strong Interval-Valued Intuitionistic Fuzzy Graph. Akram et al. (2013) studied the certain types of interval-valued fuzzy graphs. Peng Xu et al. (2022) studied the concept of certain interval-valued intuitionistic fuzzy graphs and its applications. Xindong et al. (2016) introduced the concept of interval-valued Pythagorean fuzzy set. Mohamed et. al. (2018) introduced and studied interval-valued Pythagorean fuzzy graphs. Definition 2.8 (Mohamed et. al., 2018) An Interval- valued Pythagorean Fuzzy set (IVPFS) A defined in a finite universe of discourse 𝑋 is given by 𝐴 = {〈𝑥, 𝜇𝐴(𝑥) = [ 𝜇𝐴 −(𝑥), 𝜇𝐴 +(𝑥)], 𝜂𝐴(𝑥) = [𝜂𝐴 −(𝑥), 𝜂𝐴 +(𝑥)]〉 / 𝑥𝜖 𝑋} where 𝜇𝐴 −(𝑥), 𝜇𝐴 +(𝑥) ∶ 𝑋  [0,1] and 𝜂𝐴 −(𝑥), 𝜂𝐴 +(𝑥): 𝑋  [0,1] and 0 ≤ (𝜇𝐴 +(𝑥))2 + (𝜂𝐴 +(𝑥))2 ≤ 1 . Here 𝜇𝐴(𝑥) and 𝜂𝐴(𝑥) denote the degree of membership and degree of non-membership of 𝑥 ∈ 𝑋 in 𝐴. Definition 2.9 (Mohamed et. al., 2018) An Pythagorean Fuzzy Graph (PFG) with underlying set 𝑉 defined to be a pair 𝐺 = (𝐴, 𝐵)where  the functions 𝜇𝐴: 𝑉 → 𝐷[0,1]; 𝜂𝐴: 𝑉 → 𝐷[0,1] denote the degree of membership and non-membership of the element 𝑥 ∈ 𝑉 respectively, such that 0 ≤ 𝜇𝐴(𝑥) + 𝜂𝐴(𝑥) ≤ 1 , ∀ 𝑥 ∈ 𝑉  the functions 𝜇𝐵 : 𝐸 ⊆ 𝑉 × 𝑉 → 𝐷[0,1] ; 𝜂𝐵 : 𝐸 ⊆ 𝑉 × 𝑉 → 𝐷[0,1] are defined by 𝜇𝐵 −((𝑥, 𝑦)) ≤ min( 𝜇𝐴 −(𝑥), 𝜇𝐴 −(𝑦)) ; 𝜂𝐵 −((𝑥, 𝑦)) ≥ min( 𝜂𝐴 −(𝑥), 𝜂𝐴 −(𝑦)) 𝜇𝐵 +((𝑥, 𝑦)) ≤ min( µ 𝐴 +(𝑥), µ 𝐴 +(𝑦)) ; 𝜂𝐵 +((𝑥, 𝑦)) ≥ min( 𝜂 𝐴 +(𝑥), 𝜂 𝐴 +(𝑦)) such that 0 ≤ 𝜇𝐵 +((𝑥, 𝑦)) 2 + 𝜂𝐵 +((𝑥, 𝑦)) 2 ≤ 1 , ∀(𝑥, 𝑦) ∈ 𝐸 Said Broumi et al./Decis. Mak. Appl. Manag. Eng. 5 (2) (2022) 176-200 182 Yahya et al. (2018) defined the strong interval-valued Pythagorean fuzzy graph and Cartesian product, composition and join of two strong interval-valued Pythagorean fuzzy graph are studied. Definition 2.10 (Broumi et al., 2016d) An Interval-valued Neutrosophic Set (IVNS) 𝐴 in 𝑋 is characterized by truth- membership function 𝑇𝐴(𝑥) , indeterminacy-membership function 𝐼𝐴 (𝑥) and falsity- membership function 𝐹𝐴(𝑥). For each point 𝑥 ∈ 𝑋, we have that 𝑇𝐴(𝑥) = [𝑇𝐴 −(𝑥), 𝑇𝐴 +(𝑥)], 𝐼𝐴(𝑥) = [𝐼𝐴 −(𝑥), 𝐼𝐴 +(𝑥)], 𝐹𝐴(𝑥) = [𝐹𝐴 −(𝑥), 𝐹𝐴 +(𝑥)] ⊆ [0, 1] and 0 ≤ 𝑇𝐴 (𝑥) + 𝐼𝐴(𝑥) + 𝐹𝐴(𝑥) ≤ 3. Definition 2.11 (Broumi et al., 2016d) An Interval- valued Neutrosophic Graph (IVNG) of a graph 𝐺 ∗ = (𝑉, 𝐸) we mean a pair 𝐺 = (𝐴, 𝐵), where 𝐴 =< [𝑇𝐴 −, 𝑇𝐴 +], [𝐼𝐴 −, 𝐼𝐴 +], [𝐹𝐴 −, 𝐹𝐴 +] > is an interval- valued neutrosophic set on V; and 𝐵 = 〈[𝑇𝐵 −, 𝑇𝐵 +], [𝐼𝐵 −, 𝐼𝐵 +], [𝐹𝐵 −, 𝐹𝐵 +]〉 is an interval valued neutrosophic relation on 𝐸 satisfying the following condition: i. 𝑉 = { 𝑣1 , 𝑣2 , … , 𝑣𝑛 }, such that 𝑇𝐴 − ∶ 𝑉 → [0, 1], 𝑇𝐴 + ∶ 𝑉 → [0, 1], 𝐼𝐴 − ∶ 𝑉 → [0, 1], 𝐼𝐴 +: 𝑉 → [0, 1] and 𝐹𝐴 −: 𝑉 → [0, 1], 𝐹𝐴 +: 𝑉 → [0, 1] denote the degree of truth-membership, the degree of indeterminacy-membership and falsity- membership of the element 𝑦 ∈ 𝑉, respectively, and 0 ≤ 𝑇𝐴(𝑣𝑖 ) + 𝐼𝐴 (𝑣𝑖 ) + 𝐹𝐴(𝑣𝑖 ) ≤ 3, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑖 ∈ 𝑉 (𝑖 = 1, 2, … , 𝑛) ii. The functions 𝑇𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝑇𝐵 +: 𝑉 × 𝑉 → [0, 1], 𝐼𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝐼𝐵 +: 𝑉 × 𝑉 → [0, 1] and 𝐹𝐵 −: 𝑉 × 𝑉 → [0,1], 𝐹𝐵 +: 𝑉 × 𝑉 → [0, 1] are such that 𝑇𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 −(𝑣𝑖 ), 𝑇𝐴 −(𝑣𝑗 )], 𝑇𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 +(𝑣𝑖 ), 𝑇𝐴 +(𝑣𝑗 )], 𝐼𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐼𝐵 − (𝑣𝑖 ), 𝐼𝐵 −(𝑣𝑗 )], 𝐼𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐼𝐵 +(𝑣𝑖 ), 𝐼𝐵 +(𝑣𝑗 )], 𝐹𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐹𝐵 −(𝑣𝑖 ), 𝐹𝐵 −(𝑣𝑗 )], 𝐹𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ 𝑚𝑎𝑥[𝐹𝐵 +(𝑣𝑖 ), 𝐹𝐵 +(𝑣𝑗 )], denoting the degree of truth-membership, indeterminacy-membership and falsity- membership of the edge (𝑣𝑖 , 𝑣𝑗 ) ∈ 𝐸 respectively, where 0 ≤ 𝑇𝐵 ({𝑣𝑖 , 𝑣𝑗 }) + 𝐼𝐵 ({𝑣𝑖 , 𝑣𝑗 }) + 𝐹𝐵 ({𝑣𝑖 , 𝑣𝑗 }) ≤ 3 for all {𝑣𝑖 , 𝑣𝑗 } ∈ 𝐸 (𝑖, 𝑗 = 1, 2, … , 𝑛). 3. Interval-valued Fermatean neutrosophic graphs Fuzzy sets, Intuitionistic fuzzy sets, Neutrosophic sets are the generalization of the classical set and which are also the most popular mathematical tools in the study uncertainty. Later, researchers combined these sets with graph structures and studied its properties in literature. These combinations, Fuzzy graphs, Intuitionistic fuzzy graphs and Neutrosophic graphs are useful in decision making problems. In an administrative setup, electing a leader among a group of people through the voting process, a judgement may give based on a candidate satisfies his expectations with a possibility of 0.80 and this candidate dissatisfies the expectations with a possibility of 0.95 and neutrally give 0.85 But their sum is 2.80 (>2) and their square sum is 2.265 (>2) and the sum of the cubes is equal to 1.9835 (<2). It is impossible to give an exact degree of membership in every instant, because the lack of knowledge, vague Interval- valued Fermatean Neutrosophic Graphs 183 information, and so forth may produce higher values to the membership values. To overcome this problem, we can use interval-valued fuzzy sets, which assign to each element a closed interval which approximates the “real,” but unknown, membership degree. In this series, we are adding one more class of graphs namely, interval-valued Fermatean neutrosophic graphs and certain types of interval-valued Fermatean neutrosophic graphs are introduced and discussed in this section. Definition 3.1 An interval-valued Fermatean neutrosophic set (IVFNS) 𝐴 on the universe of discourse 𝑋 is of the structure: 𝐴 = {〈𝑥, 𝑇𝐴(𝑥), IA(𝑥), FA(𝑥)〉| 𝑥 ∈ X }, where 𝑇𝐴 (𝑥) = [TA −(𝑥) , 𝑇𝐴 +(𝑥)], IA(𝑥) = [IA −(𝑥) , IA +(x)] and FA(𝑥) = [𝐹𝐴 −(𝑥), FA +(𝑥)] represents the truth-membership degree, indeterminacy-membership degree and falsity-membership degree, respectively. Consider the mapping 𝑇𝐴 (𝑥): 𝑋 → 𝐷[0,1] , IA(𝑥): 𝑋 → 𝐷[0,1], 𝐹𝐴(𝑥): 𝑋 → 𝐷[0,1] and 0 ≤ (𝑇𝐴 (𝑥)) 𝟑 +(𝐹𝐴(𝑥)) 𝟑 ≤1 and 0 ≤ (𝐼𝐴 (𝑥)) 𝟑 ≤ 1 0 ≤ (𝑇𝐴 (𝑥)) 𝟑 +(𝐹𝐴(𝑥)) 𝟑 +(𝐼𝐴 (𝑥)) 𝟑 ≤ 2 means 0 ≤ (𝑇𝐴 +(x))𝟑+(𝐼𝐴 +(𝑥))𝟑+(𝐹𝐴 +(𝑥))𝟑 ≤ 2 ∀ 𝑥 ∈ X Definition 3.2 An Interval-Valued Fermatean Neutrosophic Graph (IVFNG) of a graph 𝐺 ∗ = (𝑉, 𝐸) we mean a pair 𝐺 = (𝐴, 𝐵), where 𝐴 = 〈[𝑇𝐴 −, 𝑇𝐴 +], [𝐼𝐴 −, 𝐼𝐴 +], [𝐹𝐴 −, 𝐹𝐴 +]〉 is an interval-valued Fermatean neutrosophic set on V; and 𝐵 = 〈[𝑇𝐵 −, 𝑇𝐵 +], [𝐼𝐵 −, 𝐼𝐵 +], [𝐹𝐵 −, 𝐹𝐵 +〉] is an interval valued Fermatean neutrosophic relation on 𝐸 satisfying the following condition: i. 𝑉 = { 𝑣1 , 𝑣2 , … , 𝑣𝑛 }, such that 𝑇𝐴 − ∶ 𝑉 → [0, 1], 𝑇𝐴 + ∶ 𝑉 → [0, 1], 𝐼𝐴 − ∶ 𝑉 → [0, 1], 𝐼𝐴 +: 𝑉 → [0, 1] and 𝐹𝐴 −: 𝑉 → [0, 1], 𝐹𝐴 +: 𝑉 → [0, 1]denote the degree of truth -membership, the degree of indeterminacy-membership and falsity- membership of the element 𝑦 ∈ 𝑉, respectively, and 0 ≤ 𝑇𝐴(𝑣𝑖 ) + 𝐼𝐴(𝑣𝑖 ) + 𝐹𝐴(𝑣𝑖 ) ≤ 3, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑖 ∈ 𝑉 (𝑖 = 1, 2, … , 𝑛). ii. The functions 𝑇𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝑇𝐵 +: 𝑉 × 𝑉 → [0, 1], 𝐼𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝐼𝐵 +: 𝑉 × 𝑉 → [0, 1] and 𝐹𝐵 −: 𝑉 × 𝑉 → [0,1], 𝐹𝐵 +: 𝑉 × 𝑉 → [0, 1] are such that 𝑇𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 −(𝑣𝑖 ), 𝑇𝐴 −(𝑣𝑗 )] , 𝑇𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 +(𝑣𝑖 ), 𝑇𝐴 +(𝑣𝑗 )], 𝐼𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐼𝐵 − (𝑣𝑖 ), 𝐼𝐵 −(𝑣𝑗 )], 𝐼𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐼𝐵 +(𝑣𝑖 ), 𝐼𝐵 +(𝑣𝑗 )], 𝐹𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐹𝐵 −(𝑣𝑖 ), 𝐹𝐵 −(𝑣𝑗 )], 𝐹𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐹𝐵 +(𝑣𝑖 ), 𝐹𝐵 +(𝑣𝑗 )] denoting the degree of truth-membership, indeterminacy-membership and falsity- membership of the edge (𝑣𝑖 , 𝑣𝑗 ) ∈ 𝐸 respectively, where 0 ≤ 𝑇𝐵 ({𝑣𝑖 , 𝑣𝑗 }) 3 + 𝐼𝐵 ({𝑣𝑖 , 𝑣𝑗 }) 3 + 𝐹𝐵 ({𝑣𝑖 , 𝑣𝑗 }) 3 ≤ 2 for all {𝑣𝑖 , 𝑣𝑗 } ∈ 𝐸 (𝑖, 𝑗 = 1, 2, … , 𝑛) means 0 ≤ (𝑇𝐵 +(𝑣𝑖 , 𝑣𝑗 )) 𝟑 +(𝐼𝐵 +(𝑣𝑖 , 𝑣𝑗 )) 𝟑 +(𝐹𝐵 +(𝑣𝑖 , 𝑣𝑗 )) 𝟑 ≤ 2 ∀ 𝑥 ∈ X. Example 3.3 Consider a graph 𝐺 ∗, such that 𝑉 = {𝑥1, 𝑥2, 𝑥3}, 𝐸 = {𝑥1𝑥2, 𝑥2𝑥3, 𝑥3𝑥4, 𝑥4𝑥1}. Let 𝐴 be an interval valued Fermatean neutrosophic subset of 𝑉 and 𝐵 be an interval valued Fermatean neutrosophic subset of 𝐸, denoted by 𝐴 = { 〈𝑥1[0.85,0.95], [0.90,0.95], [0.85,0.85]〉, 〈𝑥2, [0.85,0.90], [0.90,0.95], [0.85,0.90]〉, 〈𝑥3, [0.85,0.95], [0.95,0.95], [0.85,0.95]〉 } 𝐵 = { 〈𝑥1𝑥2, [0.80,0.90], [0.90,0.95], [0.80,0.85]〉, 〈𝑥2𝑥3, [0.85,0.90], [0.90,0.95], [0.85,0.85]〉, 〈𝑥3𝑥1, [0.85,0.95], [0.90,0.95], [0.85,0.85]〉 } Said Broumi et al./Decis. Mak. Appl. Manag. Eng. 5 (2) (2022) 176-200 184 Figure 3. Interval-valued Fermatean Neutrosophic Graph G Definition 3.4. Let 𝐺 = (𝐴, 𝐵) be an IVFNG. 𝐺 is an interval valued regular Fermatean neutrosophic graph if it satisfies the following conditions: ∑ 𝑇𝐵 −(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑣1≠𝑣2 ; ∑ 𝑇𝐵 +(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑣1≠𝑣2 ∑ 𝐼𝐵 −(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑣1≠𝑣2 ; ∑ 𝐼𝐵 +(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑣1≠𝑣2 ∑ 𝐹𝐵 −(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑣1≠𝑣2 ; ∑ 𝐹𝐵 +(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑣1≠𝑣2 Definition 3.5. Let G = (A, B) be an IVFNG. G is an interval valued regular strong neutrosophic graph if it satisfies the following conditions 𝑇𝐵 −(𝑣1, 𝑣2) = min(𝑇𝐴 −(𝑣1), 𝑇𝐴 −(𝑣2)) ; ∑ 𝑇𝐵 −(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑣1≠𝑣2 𝑇𝐵 +(𝑣1, 𝑣2) = min(𝑇𝐴 +(𝑣1), 𝑇𝐴 +(𝑣2)) ; ∑ 𝑇𝐵 +(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑣1≠𝑣2 𝐼𝐵 −(𝑣1, 𝑣2) = max(𝐼𝐴 −(𝑣1), 𝐼𝐴 −(𝑣2)) ; ∑ 𝐼𝐵 −(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑣1≠𝑣2 𝐼𝐵 +(𝑣1, 𝑣2) = max(𝐼𝐴 +(𝑣1), 𝐼𝐴 +(𝑣2)) ; ∑ 𝐼𝐵 +(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑣1≠𝑣2 𝐹𝐵 −(𝑣1, 𝑣2) = max(𝐹𝐴 −(𝑣1), 𝐹𝐴 −(𝑣2)) ; ∑ 𝐹𝐵 −(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑣1≠𝑣2 𝐹𝐵 +(𝑣1, 𝑣2) = max(𝐹𝐴 +(𝑣1), 𝐹𝐴 +(𝑣2)) ; ∑ 𝐹𝐵 +(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑣1≠𝑣2 Definition 3.6. Let G = (A, B) be an IVFNG. G is a strong interval valued regular strong neutrosophic graph if it satisfies the following conditions: 𝑇𝐵 −(𝑣1, 𝑣2) = min(𝑇𝐴 −(𝑣1), 𝑇𝐴 −(𝑣2)) ; 𝐼𝐵 −(𝑣1, 𝑣2) = max(𝐼𝐴 −(𝑣1), 𝐼𝐴 −(𝑣2)) ; 𝐹𝐵 −(𝑣1, 𝑣2) = max(𝐹𝐴 −(𝑣1), 𝐹𝐴 −(𝑣2)); 𝑇𝐵 +(𝑣1, 𝑣2) = min(𝑇𝐴 +(𝑣1), 𝑇𝐴 +(𝑣2)) ; 𝐼𝐵 +(𝑣1, 𝑣2) = max(𝐼𝐴 +(𝑣1), 𝐼𝐴 +(𝑣2)) ; 𝐹𝐵 +(𝑣1, 𝑣2) = max(𝐹𝐴 +(𝑣1), 𝐹𝐴 +(𝑣2)) ; Interval- valued Fermatean Neutrosophic Graphs 185 such that 0≤ 𝑇𝐵 +(𝑣1, 𝑣2 )) + I𝐵 + (𝑣1, 𝑣2)) + 𝐹𝐵 +(𝑣1, 𝑣2)) ≤ 3, for all 𝑣1, 𝑣2 ∈ 𝐸 and 0 ≤ (𝑇𝐵 +(𝑣𝑖 , 𝑣𝑗)) 𝟑 +(𝐼𝐵 +(𝑣𝑖 , 𝑣𝑗)) 𝟑 +(𝐹𝐵 +(𝑣𝑖 , 𝑣𝑗)) 𝟑 ≤ 2 ∀ 𝑥 ∈ X Example 3.7. Let 𝐺 = (𝐴, 𝐵)be an Interval-valued Fermatean Neutrosophic graph with𝑉 = {𝑥1, 𝑥2, 𝑥3}. 𝐴 = { 〈𝑥1, [0.85,0.95], [0.90,0.95], [0.85,0.85]〉, 〈𝑥2, [0.85,0.90], [0.90,0.95], [0.85,0.90]〉, 〈𝑥3, [0.85,0.95], [0.95,0.95], [0.85,0.95]〉, }, 𝐵 = { 〈𝑥1𝑥2, [0.85,0.90], [0.90,0.95], [0.80,0.90]〉, 〈𝑥2𝑥3, [0.85,0.90], [0.95,0.95], [0.85,0.95]〉, 〈𝑥1𝑥3, [0.85,0.95], [0.95,0.95], [0.85,0.95]〉, }, Figure 4. Strong Interval-valued Fermatean Neutrosophic Graph G Definition 3.8. Let 𝐴1 and 𝐴2 be interval-valued neutrosophic subsets of 𝑉1 and 𝑉2 respectively. Let 𝐵1 and 𝐵2 interval-valued neutrosophic subsets of 𝐸1 and 𝐸2 respectively. The Cartesian product of two IVFNGs 𝐺1 and 𝐺2 is denoted by 𝐺1 × 𝐺2 = (𝐴1 × 𝐴2 , 𝐵1 × 𝐵2) and is defined as follows: i. (𝑇𝐴1 − × 𝑇𝐴2 − )(𝑥1, 𝑥2) = min (𝑇𝐴1 − (𝑥1), 𝑇𝐴2 − (𝑥2)) (𝑇𝐴1 + × 𝑇𝐴2 + )(𝑥1, 𝑥2) = min (𝑇𝐴1 + (𝑥1), 𝑇𝐴2 + (𝑥2)) (𝐼𝐴1 − × 𝐼𝐴2 − )(𝑥1, 𝑥2) = max (𝐼𝐴1 − (𝑥1), 𝐼𝐴2 − (𝑥2)) (𝐼𝐴1 + × 𝐼𝐴2 + )(𝑥1 , 𝑥2) = max (𝐼𝐴1 + (𝑥1), 𝐼𝐴2 + (𝑥2)) (𝐹𝐴1 − × 𝐹𝐴2 − )(𝑥1, 𝑥2) = max (𝐹𝐴1 − (𝑥1 ), 𝐹𝐴2 − (𝑥2)) (𝐹𝐴1 + × 𝐹𝐴2 + )(𝑥1, 𝑥2) = max (𝐹𝐴1 + (𝑥1), 𝐹𝐴2 + (𝑥2)) 𝑓𝑜𝑟 𝑎𝑙𝑙 ( 𝑥1, 𝑥2) ∈ 𝑉 ii. (𝑇𝐵1 − × 𝑇𝐵2 − )((𝑥, 𝑥2)(𝑥, 𝑦2)) = min (𝑇𝐴1 − (𝑥), 𝑇𝐵1 − (𝑥2𝑦2 )) (𝑇𝐵1 + × 𝑇𝐵2 + )((𝑥, 𝑥2)(𝑥, 𝑦2)) = min (𝑇𝐴1 + (𝑥), 𝑇𝐵1 + (𝑥2𝑦2)) (𝐼𝐵1 − × 𝐼𝐵2 − )((𝑥, 𝑥2)(𝑥, 𝑦2)) = max (𝐼𝐴1 − (𝑥), 𝐼𝐵2 − (𝑥2𝑦2)) (𝐼𝐵1 + × 𝐼𝐵2 + )((𝑥, 𝑥2)(𝑥, 𝑦2)) = max (𝐼𝐴1 + (𝑥), 𝐼𝐵2 + (𝑥2𝑦2)) (𝐹𝐵1 − × 𝐹𝐵2 − )((𝑥, 𝑥2)(𝑥, 𝑦2 )) = max (𝐹𝐴1 − (𝑥), 𝐹𝐵2 − (𝑥2𝑦2)) (𝐹𝐵1 + × 𝐹𝐵2 + )((𝑥, 𝑥2 )(𝑥, 𝑦2 )) = max (𝐹𝐴1 + (𝑥), 𝐹𝐵2 + (𝑥2𝑦2 )) ∀ 𝑥 ∈ 𝑉1 𝑎𝑛𝑑 ∀ 𝑥2𝑦2 ∈ 𝐸2 iii. (𝑇𝐵1 − × 𝑇𝐵2 − ) ((𝑥1, 𝑧) (𝑦1, 𝑧)) = min (𝑇𝐵1 − (𝑥1𝑦1), 𝑇𝐴2 − (𝑧)) Said Broumi et al./Decis. Mak. Appl. Manag. Eng. 5 (2) (2022) 176-200 186 (𝑇𝐵1 + × 𝑇𝐵2 + ) ((𝑥1, 𝑧) (𝑦1, 𝑧)) = min (𝑇𝐵1 + (𝑥1𝑦1), 𝑇𝐴2 + (𝑧)) (𝐼𝐵1 − × 𝐼𝐵2 − )((𝑥1, 𝑧) (𝑦1, 𝑧)) = max (𝐼𝐵1 − (𝑥1𝑦1), 𝐼𝐴2 − (𝑧)) (𝐼𝐵1 + × 𝐼𝐵2 + )((𝑥1, 𝑧) (𝑦1, 𝑧)) = max (𝐼𝐵1 + (𝑥1𝑦1), 𝐼𝐴2 + (𝑧)) (𝐹𝐵1 − × 𝐹𝐵2 − )((𝑥1, 𝑧) (𝑦1 , 𝑧)) = max (𝐹𝐵1 − (𝑥1𝑦1 ), 𝐹𝐴2 − (𝑧)) (𝐹𝐵1 + × 𝐹𝐵2 + )((𝑥1, 𝑧)(𝑦1, 𝑧)) = max (𝐹𝐵1 + (𝑥1𝑦1 ), 𝐹𝐴2 + (𝑧)) ∀ 𝑧 ∈ 𝑉2 𝑎𝑛𝑑 ∀ 𝑥1𝑦1 ∈ 𝐸1 Example 3.9. Let 𝐺1 ∗ = (𝐴1, 𝐵1)and 𝐺2 ∗ = (𝐴2, 𝐵2 ) be two graphs where 𝑉1 = {𝑢1, 𝑢2}, 𝑉2 = {𝑣1, 𝑣2}. Consider two interval valued Fermatean neutrosophic graphs: 𝐴1 = {〈u1, [0.85,0.95], [0.95,0.95], [0.95,0.95]〉, 〈𝑢2, [0.90,0.90], [0.95,0.95], [0.85,0.85]〉, }, 𝐵1 = {〈𝑢1𝑢2, [0.85,0.90], [0.95,0.95], [0.95,0.95]〉} ; 𝐴2 = {〈𝑣1, [0.80,0.90], [0.85,0.95], [0.95,0.85]〉, 〈𝑣2, [0.95,0.90], [0.95,0.95], [0.80,0.85]〉, }, 𝐵2 = {〈𝑣1𝑣2, [0.80,0.90], [0.95,0.95], [0.95,0.85]〉}. Figure 6. Cartesian product of two IVFNGs 𝐺1 × G2 Definition 3.10. Let 𝐺 $ = 𝐺1 $ × 𝐺2 $ = (𝑉1 × 𝑉2, 𝐸) be the composition of two graphs where 𝐸 = {(𝑥, 𝑥2) (𝑥, 𝑦2 ) /𝑥 ∈ 𝑉1, 𝑥2𝑦2 ∈ 𝐸2} ∪ {(𝑥1, 𝑧) (𝑦1, 𝑧) /𝑧 ∈ 𝑉2, 𝑥1𝑦1 ∈ 𝐸1} ∪ {( 𝑥1, 𝑥2) ( 𝑦1 , 𝑦2) |𝑥1𝑦1 ∈ 𝐸1, 𝑥2 ≠ 𝑦2 }, then the composition of interval valued Fermatean neutrosophic graphs 𝐺1[ 𝐺2] = (𝐴1 ∘ 𝐴2, 𝐵1 ∘ 𝐵2) is an interval valued Fermatean neutrosophic graphs defined by: i. (𝑇𝐴1 − ∘ 𝑇𝐴2 − ) (𝑥1, 𝑥2) = min (𝑇𝐴1 − (𝑥1), 𝑇𝐴2 − (𝑥2)) (𝑇𝐴1 + ∘ 𝑇𝐴1 + ) (𝑥1, 𝑥2) = min (𝑇𝐴1 + (𝑥1), 𝑇𝐴1 + (𝑥2)) (𝐼𝐴1 − ∘ 𝐼𝐴2 − )(𝑥1, 𝑥2) = max (𝐼𝐴1 − (𝑥1), 𝐼𝐴2 − (𝑥2)) (𝐼𝐴1 + ∘ 𝐼𝐴2 + )(𝑥1, 𝑥2) = max (𝐼𝐴1 + (𝑥1), 𝐼𝐴2 + (𝑥2)) (𝐹𝐴1 − ∘ 𝐹𝐴2 − )(𝑥1, 𝑥2) = max (𝐹𝐴1 − (𝑥1), 𝐹𝐴2 − (𝑥2)) (𝐹𝐴1 + ∘ 𝐹𝐴2 + ) (𝑥1, 𝑥2) = max (𝐹𝐴1 + (𝑥1), 𝐹𝐴2 + (𝑥2)) ∀ 𝑥1 ∈ 𝑉1, 𝑥2 ∈ 𝑉2 ii. (𝑇𝐴1 − ∘ 𝑇𝐴2 − )((𝑥, 𝑥2)(𝑥, 𝑦2)) = min (𝑇𝐴1 − (𝑥), 𝑇𝐵2 − (𝑥2𝑦2 )) Figure 5. Interval − valued Fermatean Neutrosophic Graphs G1 , G2 Interval- valued Fermatean Neutrosophic Graphs 187 (𝑇𝐴1 + ∘ 𝑇𝐴1 + )((𝑥, 𝑥2)(𝑥, 𝑦2)) = min (𝑇𝐴1 + (𝑥), 𝑇𝐵2 + (𝑥2𝑦2)) (𝐼𝐴1 − ∘ 𝐼𝐴2 − ) ((𝑥, 𝑥2)(𝑥, 𝑦2)) = max (𝐼𝐴1 − (𝑥), 𝐼𝐵2 − (𝑥2𝑦2)) (𝐼𝐴1 + ∘ 𝐼𝐴2 + ) ((𝑥, 𝑥2)(𝑥, 𝑦2 )) = max (𝐼𝐴1 + (𝑥), 𝐼𝐵2 + (𝑥2𝑦2)) (𝐹𝐴1 − ∘ 𝐹𝐴2 − )((𝑥, 𝑥2)(𝑥, 𝑦2)) = max (𝐹𝐴1 − (𝑥), 𝐹𝐵2 − (𝑥2𝑦2)) (𝐹𝐴1 + ∘ 𝐹𝐴2 + ) ((𝑥, 𝑥2)(𝑥, 𝑦2)) = max (𝐹𝐴1 + (𝑥), 𝐹𝐵2 + (𝑥2𝑦2)) ∀ 𝑥 ∈ 𝑉1, ∀𝑥2 𝑦2 ∈ 𝐸2 iii. (𝑇𝐵1 − ∘ 𝑇𝐵2 − ) ((𝑥1, 𝑧) (𝑦1, 𝑧)) = min (𝑇𝐵1 − (𝑥1𝑦1), 𝑇𝐴 2 −(𝑧)) (𝑇𝐵1 + ∘ 𝑇𝐵2 + ) ((𝑥1, 𝑧) (𝑦1, 𝑧)) = min (𝑇𝐵1 + (𝑥1𝑦1 ), 𝑇𝐴 2 +(𝑧)) (𝐼𝐵1 − ∘ 𝐼𝐵2 − ) ((𝑥1, 𝑧) (𝑦1, 𝑧)) = max (𝐼𝐵1 − (𝑥1𝑦1), 𝐼𝐴2 − (𝑧)) (𝐼𝐵1 + ∘ 𝐼𝐵2 + ) ((𝑥1, 𝑧) (𝑦1, 𝑧)) = max (𝐼𝐵1 + (𝑥1𝑦1), 𝐼𝐴2 + (𝑧)) (𝐹𝐵1 − ∘ 𝐹𝐵2 − ) ((𝑥1, 𝑧) (𝑦1, 𝑧)) = max (𝐹𝐵1 − (𝑥1𝑦1 ), 𝐹𝐴2 − (𝑧)) (𝐹𝐵1 + ∘ 𝐹𝐵2 + ) ((𝑥1, 𝑧) (𝑦1, 𝑧)) = max (𝐹𝐵1 + (𝑥1𝑦1), 𝐹𝐴2 + (𝑧)) ∀ 𝑧 ∈ 𝑉2, ∀𝑥1 𝑦1 ∈ 𝐸1 ; iv. (𝑇𝐵1 − ∘ 𝑇𝐵2 − ) ((𝑥1, 𝑥2) (𝑦1, 𝑦2)) = min (𝑇𝐴2 − (𝑥2), 𝑇𝐴2 − (𝑦2), 𝑇𝐵1 − (𝑥1𝑦1)) (𝑇𝐵1 + ∘ 𝑇𝐵2 + )((𝑥1, 𝑥2)(𝑦1 , 𝑦2)) = min (𝑇𝐴2 + (𝑥2), 𝑇𝐴2 + (𝑦2), 𝑇𝐵1 + (𝑥1𝑦1)) (𝐼𝐵1 − ∘ 𝐼𝐵2 − )((𝑥1, 𝑥2)(𝑦1, 𝑦2)) = max (𝐼𝐴2 − (𝑥2), 𝐼𝐴2 − (𝑦2), 𝐼𝐵1 − (𝑥1𝑦1)) (𝐼𝐵1 + ∘ 𝐼𝐵2 + )((𝑥1, 𝑥2)(𝑦1, 𝑦2)) = max (𝐼𝐴2 + (𝑥2), 𝐼𝐴2 + (𝑦2), 𝐼𝐵1 + (𝑥1𝑦1)) (𝐹𝐵1 − ∘ 𝐹𝐵2 − )((𝑥1, 𝑥2)(𝑦1, 𝑦2 )) = max (𝐹𝐴2 − (𝑥2), 𝐹𝐴2 − (𝑦2), 𝐹𝐵1 − (𝑥1𝑦1)) ( 𝐹𝐵1 + ∘ 𝐹𝐵2 + )((𝑥1, 𝑥2)(𝑦1 , 𝑦2)) = max ( 𝐹𝐴2 + (𝑥2 ), 𝐹𝐴2 + (𝑦2 ), 𝐹𝐵1 + (𝑥1𝑦1 )), ∀ (𝑥1, 𝑥2)( 𝑦1, 𝑦2) ∈ 𝐸 0 − 𝐸, 𝑤ℎ𝑒𝑟𝑒 𝐸0 = 𝐸 ∪ {( 𝑥1, 𝑥2) ( 𝑦1, 𝑦2) |𝑥1 𝑦1 ∈ 𝐸1, 𝑥2 ≠ 𝑦2 }. Example 3.11. Let 𝐺1 ∗ = (𝐴1, 𝐵1)and 𝐺2 ∗ = (𝐴2, 𝐵2 ) be two graphs where 𝑉1 = {𝑢1, 𝑢2}, 𝑉2 = {𝑣1, 𝑣2}. Consider two interval valued Fermatean neutrosophic graphs: 𝐴1 = {〈u1, [0.85,0.95], [0.95,0.95], [0.95,0.95]〉, 〈u2, [0.90,0.90], [0.95,0.95], [0.85,0.85]〉}, 𝐵1 = {〈𝑢1𝑢2, [0.85,0.90], [0.95,0.95], [0.95,0.95]〉 } ; 𝐴2 = {〈𝑣1, [0.80,0.90], [0.85,0.95], [0.95,0.85]〉, 〈𝑣2, [0.95,0.90], [0.95,0.95], [0.80,0.85]〉}, 𝐵2 = {〈𝑣1𝑣2, [0.80,0.90], [0.95,0.95], [0.95,0.85]〉}. Figure 7. Interval − valued Fermatean Neutrosophic Graphs G1 , G2 Said Broumi et al./Decis. Mak. Appl. Manag. Eng. 5 (2) (2022) 176-200 188 Figure 8. Composition of interval valued Fermatean neutrosophic graphs 𝐺1[ 𝐺2] Definition 3.12. The union 𝐺1 ∪ 𝐺2 = (𝐴1 ∪ 𝐴2, 𝐵1 ∪ 𝐵2) of two interval valued Fermatean neutrosophic graphs of the graphs 𝐺1 ∗ and 𝐺2 ∗ is an interval-valued Fermatean neutrosophic graph of 𝐺1 ∗ ∪ 𝐺2 ∗ .  (𝑇𝐴1 − ∪ 𝑇𝐴2 − )(𝑥) = { 𝑇𝐴1 − (𝑥), 𝑖𝑓 𝑥 ∈ 𝑉1 𝑎𝑛𝑑 𝑥 ∉ 𝑉2 𝑇𝐴2 − (𝑥)𝑖𝑓 𝑥 ∉ 𝑉1 𝑎𝑛𝑑 𝑥 ∈ 𝑉2 min (𝑇𝐴1 − (𝑥), 𝑇𝐴2 − (𝑥)) if 𝑥 ∈ V1 ∩ V2, ,  (𝑇𝐴1 + ∪ 𝑇𝐴2 + )(𝑥) = { 𝑇𝐴1 + (𝑥), 𝑖𝑓 𝑥 ∈ 𝑉1 𝑎𝑛𝑑 𝑥 ∉ 𝑉2 𝑇𝐴2 + (𝑥)𝑖𝑓 𝑥 ∉ 𝑉1 𝑎𝑛𝑑 𝑥 ∈ 𝑉2 min (𝑇𝐴1 + (𝑥), 𝑇𝐴2 + (𝑥)) if 𝑥 ∈ V1 ∩ V2,  (𝐼𝐴1 − ∪ 𝐼𝐴2 − )(𝑥) = { 𝐼𝐴1 − (𝑥), 𝑖𝑓 𝑥 ∈ 𝑉1 𝑎𝑛𝑑 𝑥 ∉ 𝑉2 𝐼𝐴2 − (𝑥)𝑖𝑓 𝑥 ∉ 𝑉1 𝑎𝑛𝑑 𝑥 ∈ 𝑉2 max (𝐼𝐴1 − (𝑥), 𝐼𝐴2 − (𝑥)) if 𝑥 ∈ V1 ∩ V2,  (𝐼𝐴1 + ∪ 𝐼𝐴2 + )(𝑥) = { 𝐼𝐴1 + (𝑥), 𝑖𝑓 𝑥 ∈ 𝑉1 𝑎𝑛𝑑 𝑥 ∉ 𝑉2 𝐼𝐴2 + (𝑥)𝑖𝑓 𝑥 ∉ 𝑉1 𝑎𝑛𝑑 𝑥 ∈ 𝑉2 max (𝐼𝐴1 + (𝑥), 𝐼𝐴2 + (𝑥)) if 𝑥 ∈ V1 ∩ V2,  (𝐹𝐴1 − ∪ 𝐹𝐴2 − )(𝑥) = { 𝐹𝐴1 − (𝑥), 𝑖𝑓 𝑥 ∈ 𝑉1 𝑎𝑛𝑑 𝑥 ∉ 𝑉2 𝐹𝐴2 − (𝑥)𝑖𝑓 𝑥 ∉ 𝑉1 𝑎𝑛𝑑 𝑥 ∈ 𝑉2 max (𝐹𝐴1 − (𝑥), 𝐹𝐴2 − (𝑥)) if 𝑥 ∈ V1 ∩ V2,  (𝐹𝐴1 + ∪ 𝐹𝐴2 + )(𝑥) = { 𝐹𝐴1 + (𝑥), 𝑖𝑓 𝑥 ∈ 𝑉1 𝑎𝑛𝑑 𝑥 ∉ 𝑉2 𝐹𝐴2 + (𝑥)𝑖𝑓 𝑥 ∉ 𝑉1 𝑎𝑛𝑑 𝑥 ∈ 𝑉2 max (𝐹𝐴1 + (𝑥), 𝐹𝐴2 + (𝑥)) if 𝑥 ∈ V1 ∩ V2,  (𝑇𝐵1 − ∪ 𝑇𝐵2 − )(𝑥𝑦) = { 𝑇𝐵1 − (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∉ 𝐸2 𝑇𝐵2 − (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∉ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∈ 𝐸2 min (𝑇𝐵1 − (𝑥𝑦), 𝑇𝐵2 − (𝑥𝑦) ) if 𝑥𝑦 ∈ E1 ∩ 𝐸2,  (𝑇𝐵1 + ∪ 𝑇𝐵2 + )(𝑥𝑦) = { 𝑇𝐵1 + (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∉ 𝐸2 𝑇𝐵2 + (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∉ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∈ 𝐸2 min (𝑇𝐵1 + (𝑥𝑦), 𝑇𝐵2 + (𝑥𝑦) ) if 𝑥𝑦 ∈ E1 ∩ 𝐸2, Interval- valued Fermatean Neutrosophic Graphs 189  (𝐼𝐵1 − ∪ 𝐼𝐵2 − )(𝑥𝑦) = { 𝐼𝐵1 − (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∉ 𝐸2 𝐼𝐵2 − (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∉ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∈ 𝐸2 min (𝐼𝐵1 − (𝑥𝑦), 𝐼𝐵2 − (𝑥𝑦) ) if 𝑥𝑦 ∈ E1 ∩ 𝐸2,  (𝐼𝐵1 + ∪ 𝐼𝐵2 + )(𝑥𝑦) = { 𝐼𝐵1 + (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∉ 𝐸2 𝐼𝐵2 + (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∉ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∈ 𝐸2 max (𝐼𝐵1 + (𝑥𝑦), 𝐼𝐵2 + (𝑥𝑦) ) if 𝑥𝑦 ∈ E1 ∩ 𝐸2,  𝐹𝐵1 − ∪ 𝐹𝐵2 − )(𝑥𝑦) = { 𝐹𝐵1 − (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∉ 𝐸2 𝐹𝐵2 − (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∉ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∈ 𝐸2 max (𝐹𝐵1 − (𝑥𝑦), 𝐹𝐵2 − (𝑥𝑦) ) if 𝑥𝑦 ∈ E1 ∩ 𝐸2,  (𝐹𝐵1 + ∪ 𝐹𝐵2 + )(𝑥𝑦) = { 𝐹𝐵1 + (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∉ 𝐸2 𝐹𝐵2 + (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∉ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∈ 𝐸2 max (𝐹𝐵1 + (𝑥𝑦), 𝐹𝐵2 + (𝑥𝑦) ) if 𝑥𝑦 ∈ E1 ∩ 𝐸2, Definition 3.13. The join of 𝐺1 + 𝐺2 = (𝐴1 + 𝐴2 , 𝐵1 + 𝐵2 ) interval valued neutrosophic graphs 𝐺1 and 𝐺2 of the graphs 𝐺1 ∗ and 𝐺2 ∗ is defined as follows:  (𝑇𝐴1 − + 𝑇𝐴2 − )(𝑥) = { 𝑇𝐴1 − (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉1 𝑇𝐴2 − (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉2 min(𝑇𝐴1 − , 𝑇𝐴2 − )(𝑥) if 𝑥 ∈ V1 ∪ V2, ,  (𝑇𝐴1 + + 𝑇𝐴2 + )(𝑥) = { 𝑇𝐴1 + (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉1 𝑇𝐴2 + (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉2 min(𝑇𝐴1 + , 𝑇𝐴2 + )(𝑥) if 𝑥 ∈ V1 ∪ V2,  (𝐼𝐴1 − + 𝐼𝐴2 − )(𝑥) = { 𝐼𝐴1 − (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉1 𝐼𝐴2 − (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉2 max(𝐼𝐴1 − , 𝐼𝐴2 − )(𝑥) if 𝑥 ∈ V1 ∪ V2,  (𝐼𝐴1 + + 𝐼𝐴2 + )(𝑥) = { 𝐼𝐴1 + (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉1 𝐼𝐴2 + (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉2 max(𝐼𝐴1 + , 𝐼𝐴2 + )(𝑥) if 𝑥 ∈ V1 ∪ V2,  (𝐹𝐴1 − + 𝐹𝐴2 − )(𝑥) = { 𝐹𝐴1 − (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉1 𝐹𝐴2 − (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉2 max(𝐹𝐴1 − , 𝐹𝐴2 − )(𝑥) if 𝑥 ∈ V1 ∪ V2,  (𝐹𝐴1 + + 𝐹𝐴2 + )(𝑥) = { 𝐹𝐴1 + (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉1 𝐹𝐴2 + (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉2 max(𝐹𝐴1 + , 𝐹𝐴2 + )(𝑥) if 𝑥 ∈ V1 ∪ V2,  (𝑇𝐵1 − + 𝑇𝐵2 − )(𝑥𝑦) = { 𝑇𝐵1 − (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝑇𝐵2 − (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∈ 𝐸2 min (𝑇𝐵1 − (𝑥𝑦), 𝑇𝐵2 − (𝑥𝑦) ) if 𝑥𝑦 ∈ E1 ∪ 𝐸2, Said Broumi et al./Decis. Mak. Appl. Manag. Eng. 5 (2) (2022) 176-200 190  (𝑇𝐵1 + + 𝑇𝐵2 + )(𝑥𝑦) = { 𝑇𝐵1 + (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝑇𝐵2 + (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∈ 𝐸2 min (𝑇𝐵1 + (𝑥𝑦), 𝑇𝐵2 + (𝑥𝑦) ) if 𝑥𝑦 ∈ E1 ∪ 𝐸2,  𝐼𝐵1 − + 𝐼𝐵2 − )(𝑥𝑦) = { 𝐼𝐵1 − (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝐼𝐵2 − (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∈ 𝐸2 max(𝐼𝐵1 − (𝑥𝑦), 𝐼𝐵2 − (𝑥𝑦) ) if 𝑥𝑦 ∈ E1 ∪ 𝐸2,  (𝐼𝐵1 + + 𝐼𝐵2 + )(𝑥𝑦) = { 𝐼𝐵1 + (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝐼𝐵2 + (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∈ 𝐸2 max(𝐼𝐵1 + (𝑥𝑦), 𝐼𝐵2 + (𝑥𝑦) ) if xy ∈ E1 ∪ 𝐸2,  𝐹𝐵1 − + 𝐹𝐵2 − )(𝑥𝑦) = { 𝐹𝐵1 − (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝐹𝐵2 − (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∈ 𝐸2 max(𝐹𝐵1 − (𝑥𝑦), 𝐹𝐵2 − (𝑥𝑦) ) if 𝑥𝑦 ∈ E1 ∪ 𝐸2,  (𝐹𝐵1 + + 𝐹𝐵2 + )(𝑥𝑦) = { 𝐹𝐵1 + (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝐹𝐵2 + (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∈ 𝐸2 max(𝐹𝐵1 + (𝑥𝑦), 𝐹𝐵2 + (𝑥𝑦) ) if 𝑥𝑦 ∈ E1 ∪ 𝐸2,  (𝑇𝐵1 − + 𝑇𝐵2 − ) (𝑥 𝑦) = min (𝑇𝐵1 − (𝑥), 𝑇𝐵2 − (𝑥))  (𝑇𝐵1 + + 𝑇𝐵2 + ) (𝑥𝑦) = min (𝑇𝐵1 + (𝑥), 𝑇𝐵2 + (𝑥))  (𝐼𝐵1 − + 𝐼𝐵2 − ) (𝑥𝑦) = max (𝐼𝐵1 − (𝑥), 𝐼𝐵2 − (𝑥))  (𝐼𝐵1 + + 𝐼𝐵2 + ) (𝑥 𝑦) = max (𝐼𝐵1 + (𝑥), 𝐼𝐵2 + (𝑥)  (𝐹𝐵1 − + 𝐹𝐵2 − ) (𝑥𝑦) = max (𝐹𝐵1 − (𝑥), 𝐹𝐵2 − (𝑥))  (𝐹𝐵1 + + 𝐹𝐵2 + ) (𝑥 𝑦) = max (𝐹𝐵1 + (𝑥), 𝐹𝐵1 + (𝑥))𝑖𝑓𝑥𝑦 ∈ 𝐸′ , where 𝐸′is the set of all edges joining the nodes of 𝑉1 and 𝑉2, assuming 𝑉1 ∩ 𝑉2 = ∅. Example 3.14. Let 𝐺1 ∗ = (𝐴1, 𝐵1)and 𝐺2 ∗ = (𝐴2, 𝐵2 ) be two graphs where𝑉1 = {𝑢1, 𝑢2, 𝑢3,𝑢4}, 𝑉2 = {𝑣1, 𝑣2, 𝑣3}. Consider two interval valued fermatean neutrosophic graphs: 𝐴1 = { 〈𝑢1, [0.85,0.95], [0.95,0.95], [0.95,0.95]〉, 〈𝑢2, [0.90,0.90], [0.95,0.95], [0.85,0.85]〉, 〈𝑢3, [0.90,0.95], [0.85,0.95], [0.85,0.85]〉, 〈𝑢4, [0.90,0.95], [0.95,0.90], [0.80,0.85]〉 }, 𝐵1 = { 〈𝑢1𝑢2, [0.85,0.90], [0.95,0.95], [0.95,0.95]〉, 〈𝑢2𝑢3, [0.90,0.90], [0.95,0.95], [0.85,0.85]〉, 〈𝑢3𝑢4, [0.90,0.95], [0.95,0.95], [0.85,0.85]〉, 〈𝑢1𝑢4, [0.85,0.95], [0.95,0.95], [0.95,0.95]〉, 〈𝑢1𝑢3, [0.85,0.95], [0.95,0.95], [0.95,0.95]〉 } ; 𝐴2 = { 〈𝑢1, [0.80,0.90], [0.85,0.95], [0.95,0.85]〉, 〈𝑢2, [0.95,0.90], [0.95,0.95], [0.80,0.85]〉, 〈𝑢3, [0.90,0.90], [0.95,0.95], [0.80,0.80]〉 }, 𝐵2 = { 〈𝑢1𝑢2, [0.80,0.90], [0.95,0.95], [0.95,0.85]〉 , 〈𝑢2𝑢3, [0.90,0.90], [0.95,0.95], [0.80,0.85]〉, 〈𝑢1𝑢3, [0.80,0.90], [0.95,0.95], [0.95,0.85]〉 }. Interval- valued Fermatean Neutrosophic Graphs 191 Figure 9. Interval − valued Fermatean Neutrosophic Graph G1 Figure 10. Interval − valued Fermatean Neutrosophic Graph G2 Figure 11. Union two Interval − valued Fermatean Neutrosophic Graphs 𝐺1 ∪ 𝐺2 { 〈𝑢1𝑢2, [0.80,0.90], [0.95,0.95], [0.95,0.95]〉, 〈𝑢2𝑢3, [0.90,0.90], [0.95,0.95], [0.85,0.85]〉, 〈𝑢3𝑢4, [0.90,0.95], [0.95,0.95], [0.85,0.85]〉, 〈𝑢1𝑢4, [0.85,0.95], [0.95,0.95], [0.95,0.95]〉, 〈𝑢1𝑢3, [0.80,0.90], [0.95,0.95], [0.95,0.95]〉 } Example 3.15 Let 𝐺1 ∗ = (𝐴1, 𝐵1)and 𝐺2 ∗ = (𝐴2, 𝐵2 ) be two graphs where 𝑉1 = {𝑥1, 𝑥2, 𝑥3}, 𝑉2 = {𝑦1, 𝑦2, 𝑦3}. Consider two interval valued Fermatean neutrosophic graphs : 𝐴1 = { 〈𝑥1, [0.85,0.95], [0.95,0.95], [0.95,0.95]〉, 〈𝑥2, [0.90,0.90], [0.95,0.95], [0.85,0.85]〉, 〈𝑥3, [0.90,0.95], [0.85,0.95], [0.85,0.85]〉 }, 𝐵1 = {〈𝑥1𝑥2, [0.85,0.90], [0.95,0.95], [0.95,0.95]〉, 〈𝑥2𝑥3, [0.90,0.90], [0.95,0.95], [0.85,0.85]〉 } 𝐴2 = { 〈𝑦1, [0.85,0.85], [0.95,0.95], [0.90,0.90]〉, 〈𝑦2, [0.95,0.90], [0.90,0.95], [0.80,0.85]〉, 〈𝑦3, [0.95,0.95], [0.85,0.85], [0.85,0.85]〉 }, 𝐵2 = {〈𝑦1𝑦2, [0.85,0.85], [0.95,0.95], [0.90,0.90]〉, 〈𝑦2𝑦3, [0.95,0.90], [0.90,0.95], [0.85,0.85]〉} Said Broumi et al./Decis. Mak. Appl. Manag. Eng. 5 (2) (2022) 176-200 192 Figure 12. Interval − valued Fermatean Neutrosophic Graph G1 Figure 13. Interval − valued Fermatean Neutrosophic Graph G2 Figure 14. Join of Interval − valued Fermatean Neutrosophic Graphs 𝐺1 + 𝐺2 𝐸(𝐺1 + 𝐺2) : < 𝑥1𝑥2, [0.85,0.90], [0.95,0.95], [0.95,0.95] >, < 𝑥2𝑥3, [0.90,0.90], [0.95,0.95], [0.85,0.85] > < 𝑦1𝑦2, [0.85,0.85], [0.95,0.95], [0.90,0.90] >, < 𝑦2𝑦3, [0.95,0.90], [0.90,0.95], [0.85,0.85] > < 𝑥1𝑦1, [0.85,0.85], [0.95,0.95], [0.95,0.95] >, < 𝑥1𝑦2, [0.85,0.90], [0.95,0.95], [0.95,0.95] >, < 𝑥1𝑦3, [0.85,0.95], [0.95,0.95], [0.95,0.95] > < 𝑥2𝑦1, [0.85,0.90], [0.95,0.95], [0.90,0.90] >, < 𝑥2𝑦2, [0.90,0.90], [0.95,0.95], [0.85,0.85] >, < 𝑥2𝑦3, [0.90,0.90], [0.95,0.95], [0.85,0.85] > < 𝑥3𝑦1, [0.85,0.85], [0.95,0.95], [0.90,0.90] >, < 𝑥3𝑦2, [0.90,0.90], [0.90,0.95], [0.85,0.85] >, < 𝑥3𝑦3, [0.90,0.95], [0.85,0.95], [0.85,0.85] > Interval- valued Fermatean Neutrosophic Graphs 193 Definition 3.16. An interval valued Fermatean neutrosophic graph G = (A, B) is called complete if 𝑇𝐵 −({𝑣𝑖 , 𝑣𝑗}) = min[𝑇𝐴 −(𝑣𝑖 ), 𝑇𝐴 −(𝑣𝑗 )] , 𝑇𝐵 +({𝑣𝑖 , 𝑣𝑗 }) = min[𝑇𝐴 +(𝑣𝑖 ), 𝑇𝐴 +(𝑣𝑗 )] 𝐼𝐵 −({𝑣𝑖 , 𝑣𝑗 }) = max[𝐼𝐵 − (𝑣𝑖 ), 𝐼𝐵 −(𝑣𝑗 )] , 𝐼𝐵 +({𝑣𝑖 , 𝑣𝑗 }) = max[𝐼𝐵 +(𝑣𝑖 ), 𝐼𝐵 +(𝑣𝑗 )] 𝐹𝐵 −({𝑣𝑖 , 𝑣𝑗 }) = max[𝐹𝐵 −(𝑣𝑖 ), 𝐹𝐵 −(𝑣𝑗 )], 𝐹𝐵 +({𝑣𝑖 , 𝑣𝑗 }) = max[𝐹𝐵 +(𝑣𝑖 ), 𝐹𝐵 +(𝑣𝑗 )] Definition 3.17. Let G = (A,B) be an interval-valued Fermatean neutrosophic graph where 𝐴 = 〈[𝑇𝐴 −, 𝑇𝐴 +], [𝐼𝐴 −, 𝐼𝐴 +], [𝐹𝐴 −, 𝐹𝐴 +]〉 is an interval-valued Fermatean neutrosophic set on V; and 𝐵 = 〈[𝑇𝐵 −, 𝑇𝐵 +], [𝐼𝐵 −, 𝐼𝐵 +], [𝐹𝐵 −, 𝐹𝐵 +]〉 is an interval valued Fermatean neutrosophic relation on 𝐸 satisfying 𝑉 = { 𝑣1 , 𝑣2 , … , 𝑣𝑛 }, such that 𝑇𝐴 − ∶ 𝑉 → [0, 1], 𝑇𝐴 + ∶ 𝑉 → [0, 1], 𝐼𝐴 − ∶ 𝑉 → [0, 1], 𝐼𝐴 +: 𝑉 → [0, 1] and 𝐹𝐴 −: 𝑉 → [0, 1], 𝐹𝐴 +: 𝑉 → [0, 1] denote the degree of truth-membership, the degree of indeterminacy-membership and falsity- membership of the element 𝑦 ∈ 𝑉, respectively. The positive degree of a vertex 𝑢 ∈ 𝑉(𝐺) is 𝑇+(𝑢) = ∑ [𝑇𝐴 +]𝑢𝑣∈𝐸(𝐺) ; 𝐼 +(𝑢) = ∑ [𝐼𝐴 +]𝑢𝑣∈𝐸(𝐺) ; 𝐹 +(𝑢) = ∑ [𝐹𝐴 +]𝑢𝑣∈𝐸(𝐺) and 𝑑 +(𝑢) = (𝑇𝐴 +, 𝐼𝐴 +, 𝐹𝐴 +). 𝑇−(𝑢) = ∑ [𝑇𝐴 −]𝑢𝑣∈𝐸(𝐺) ; 𝐼 −(𝑢) = ∑ [𝐼𝐴 −]𝑢𝑣∈𝐸(𝐺) ; 𝐹 −(𝑢) = ∑ [𝐹𝐴 −]𝑢𝑣∈𝐸(𝐺) and 𝑑 −(𝑢) = (𝑇𝐴 −, 𝐼𝐴 −, 𝐹𝐴 −). The degree of a vertex 𝑢 is 𝑑(𝑢) = [𝑑+ (𝑢), 𝑑− (𝑢)]. If 𝑑+ (𝑢) = 𝑘1 , 𝑑 − (𝑢) = 𝑘2 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑢 ∈ 𝑉, 𝑘1, 𝑘2 are two real numbers, then the graph is called [𝑘1 , 𝑘2] -regular interval valued Fermatean neutrosophic graph. Example 3.18. We consider an interval-valued Fermatean neutrosophic graph. Figure 15. Interval- valued Fermatean Neutrosophic Graph G 𝑑(x1) = ([ 1.65,1.80,1.65], [1.85,1.90,1.70]); d(x2) = ([1.65,1.8,1.65], [1.8,1.9,1.7]); d(x3) = ([1.7,1.8,1.7], [1.85,1.9,1.7]). 4. Proposed IVFNG framework for MCDM problem The most of real life problems deal with uncertain domain. Recently, researchers (Sriganesh et al. 2021; Sundareswaran et al. 2022) have been studied the assessment of structural cracks in buildings using single-valued neutrosophic DEMATEL model and graph theoretical approach. The new concepts of IVFNG are employed to find the best materials that are used for making dental implants in the case of smokers. There are many researchers developed and studied different types uncertainty sets and their application in Multi-Criteria Decision- Making (MCDM) (Duran et al., 2021; Ejegwa et al. 2022; Mohanta et al., 2020; Li et al., 2022; Smarandache, 2020; Smarandache, 2022; Wang et al., 2022; Zhang et al., 2022). Mahesh et al. (2022), made a comparative study Said Broumi et al./Decis. Mak. Appl. Manag. Eng. 5 (2) (2022) 176-200 194 of Dental Implant Materials Using Digraph Techniques. Dental implants are the most popular option to replace missing teeth. They create direct contact with the bone which mimics the root of the tooth, upon which dental prosthesis can be fitted. These implants are designed in such a way that they can last for a long time without any failure. They get adhered to the bone without intervening in any connective tissue and this phenomenon is known as osseointegration. Titanium is considered the gold standard as it is the most commonly used dental implant material in use since the 1960s Zirconia is a non-metallic alternative to metal dental implants like 𝑇𝑖 𝑎𝑙𝑙𝑜𝑦 (𝑇𝑖 − 6𝐴𝑙 − 4𝑉) and 𝑇𝑖 alloys. Figure 16. Fishbone diagram with the various factors and subfactors In this section, the concept of Interval-valued Fermatean neutrosophic graph- theoretic approach has been used to selection of material. The condition of osseointegration in smokers is taken into consideration to compare the different material dental implants namely 𝑇𝑖 𝑎𝑙𝑙𝑜𝑦 (𝑇𝑖 − 6𝐴𝑙 − 4𝑉), 𝑇𝑖 alloy, and zirconia. The material to be chosen should exhibit certain properties to satisfy the purpose. While designing a dental implant, many factors come into consideration such as materials, dimensions, shape, etc. Material selection is the most important property for a dental implant to serve the required function. The material of the implant must be affordable and available. Following are the factors that are important for the selection of the material. Biocompatibility (B): A biocompatible material does not invoke an immune response and does not release any toxic substances. The major subfactors of biocompatibility are corrosion, inflammation, and allergy. Surface Properties (S): Surface properties refer to macroscopic and microscopic features of the implant surface and it plays a major role in determining the level of osseointegration between the implant and the bone. The major subfactors of surface properties are Surface Tension and Surface Energy, Surface Roughness, Porosity. Mechanical Properties(M): The implant biomaterial should possess a high degree of modulus of elasticity, to withstand the forces applied to the implant, thus preventing its deformation. It also ensures uniform stress distribution, thus reducing the implant movement concerning the bone. Cost (C): Dental implants in India range from 30,000-50,000 rupees. The price depends on many factors like the type of tooth implant, material, and design of the implant, etc. Titanium is more expensive than stainless steel. The cost of titanium is slightly lower than zirconia. Interval- valued Fermatean Neutrosophic Graphs 195 Titanium (𝑀1) and Titanium Alloys (𝑀2): Titanium is an excellent corrosion– resistant material due to the formation of 𝑇𝑖 𝑎𝑙𝑙𝑜𝑦 (𝑇𝑖 − 6𝐴𝑙 − 4𝑉) when 𝑇𝑖 atoms react with water molecules and oxygen. They show excellent biocompatibility properties and support osseointegration. Titanium-based dental implants are strong and resist fracture. The cost of titanium is slightly lower than the zirconia. However, titanium implants are less aesthetically pleasing than zirconia and hence they are not preferable to use in the case of front teeth implant placement. Zirconia could be preferred in this case due to its ivory color. Zirconia (𝑀3): Zirconia is a non-metallic alternative to metal dental implants like Ti. An advantage of zirconia over titanium is its ivory color. Its low modulus of elasticity and thermal conductivity, low affinity to plaque, and high biocompatibility, in addition to its white color, have made zirconia ceramics a very attractive alternative to titanium. It is highly corrosion resistant and does not involve any release of ions hence no cytotoxicity. Figure 17. Types of Dental Implants In the process of applying IVFNG in identifying the best material. IVFNG can be represented as a matrix whose rows and columns are the sub-factors. 𝑉 = { 𝑀1, 𝑀2, 𝑀3} be the three different material under the selection on the basis of wishing param eters or attributes set 𝐴 = {𝐵 , 𝑆}. Figure 18. IVFNG based on Biocompatibility & Surface Properties We construct the adjacency matrix for 𝑀(𝐵), 𝑀(𝑆) listed below: 𝑴(𝑩) = ( < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.95], [0.95,0.95], [0.85,0.85] > < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.95], [0.95,0.95], [0.85,0.85] > < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.85], [0.95,0.95], [0.85,0.95] > < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.85], [0.95,0.95], [0.85,0.95] > < [0, 0 ], [0, 0 ], [0, 0 ] > ) 𝑴(𝑺) = ( < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.95], [0.90,0.95], [0.85,0.85] > < [0.85,0.90], [0.95,0.95], [0.95,0.85] > < [0.85,0.95], [0.90,0.95], [0.85,0.85] > < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.90], [0.95,0.95], [0.95,0.85] > < [0.85,0.90], [0.95,0.95], [0.95,0.85] > < [0.85,0.90], [0.95,0.95], [0.95,0.85] > < [0, 0 ], [0, 0 ], [0, 0 ] > ) Said Broumi et al./Decis. Mak. Appl. Manag. Eng. 5 (2) (2022) 176-200 196 We obtain the resultant interval valued Fermatean neutrosophic graph G by performing some operation (AND or OR). The incidence matrix of resultant interval Fermatean neutrosophic graph is 𝑴(𝑩) = ( < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.95], [0.95,0.95], [0.85,0.85] > < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.95], [0.95,0.95], [0.85,0.85] > < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.85], [0.95,0.95], [0.95,0.95] > < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.85], [0.95,0.95], [0.95,0.95] > < [0, 0 ], [0, 0 ], [0, 0 ] > ) Sahin (2015) defined the average possible membership degree of element x to interval valued neutrosophic set 𝐴 = 〈[𝑇𝐴 − (𝑥), 𝑇𝐴 +(𝑥)], [𝐼𝐴 − (𝑥), 𝐼𝐴 +(𝑥)], [𝐹𝐴 − (𝑥), 𝐹𝐴 +(𝑥)]〉 as follows: 𝑆𝑘 (𝑥) = 𝑇𝐴 − (𝑥) + 𝑇𝐴 +(𝑥) + 4 − 𝐼𝐴 − (𝑥) − 𝐼𝐴 +(𝑥) − 𝐹𝐴 − (𝑥) − 𝐹𝐴 +(𝑥) 6 Based on 𝑆𝑘 (𝑥), Table 2 depicted the score value of adjacency matrix of resultant interval valued Fermatean neutrosophic graph G with 𝑆𝑘 and choice value for both materials. Table 2. Score value of adjacency matrix Materials 𝑴𝟏 𝑴𝟐 𝑴𝟑 Overall 𝑴𝟏 0 0.383 0 0.383 𝑴𝟐 0.383 0 0.317 0.7 𝑴𝟑 0 0.317 0 0.317 Further, it is noticed from Table 2, 𝑇𝑖 𝑎𝑙𝑙𝑜𝑦 (𝑇𝑖 − 6𝐴𝑙 − 4𝑉) has higher level of osseointegration in smokers followed by 𝑇𝑖 and zirconia. Therefore, we may claim that IVFNG is a new way to tackle the uncertainty in Fermatean Neutrosophic environment. 5. Conclusion The concept of uncertainty plays a vital role in all science and engineering problems. Especially, Fuzzy theory, Intuitionistic fuzzy theory and then Neutrosophic theory are the most valuable tools to find the optimum solution in mutli-criteria decision making problems. In this work, we include one more concept called interval- valued Fermatean neutrosophic graphs in the list which has Pythagorean Neutrosophic, Single Valued Neutrosophic, Bipolar Neutrosophic graphs. We have discussed various types of Interval-valued Fermatean Neutrosophic graphs and the other types of these graphs in this paper. We also apply this new type of graph in a decision making problem. We are extending our research on this new concept to introduce Interval-valued Fermatean Neutrosophic number and Interval-valued Fermatean triangle and trapezoidal Neutrosophic number and its applications in our future work. Interval-valued Fermatean Neutrosophic graph has many advantages in MCDM problems such as mobile networking, supply chain management system, bio-medical applications, e-waste management and networking, etc. In future, one may determine the optimum alternatives in MCDM problems using IVFNG based score and accuracy functions. Author Contributions: Conceptualization, S.B and F.S; methodology, S.B., M.T., and A.B; software, R.S.; validation, R.S. and M.S.; formal analysis, R.S.; investigation, S.B.; resources, S.B., R.S., and M.S.; writing—original draft preparation, R.S., and M.S.; writing—review and editing, R.S., and M.S.; visualization, R.S., and M.S.; supervision, Interval- valued Fermatean Neutrosophic Graphs 197 S.B and G.N. All authors have read and agreed to the published version of the manuscript. Funding: This research is not financially supported by any funding agencies. Data Availability Statement: Not applicable. Acknowledgments: The authors would like to thank the management of Sri Sivasubramaniya Nadar College of Engineering for giving the needed facilities to complete this research work. Conflicts of Interest: Authors declares that no competing interests for this research article. References Ajay, D., & Chellamani, P. (2020). Pythagorean Neutrosophic Fuzzy Graphs. 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