Operational Research in Engineering Sciences: Theory and Applications 
Vol. 5, Issue 3, 2022, pp. 68-91 
ISSN: 2620-1607 
eISSN: 2620-1747 

 DOI: https://doi.org/10.31181/oresta031022031b 

* Corresponding author 
broumisaid78@gmail.com, S.broumi@flbenmsik.ma (S.B) 
dajaypravin@gmail.com (A. Pravin), joshmani238@gmail.com (P. Chellamani), 
lathamax@gmail.com (L. Karthik), taleamohamed@yahoo.fr (M. Talea), assiabakali@yahoo.fr 
(A. Bakali), flippe2@gmail.com (P. Schweizer), jafaripersia@gmail.com (S. Jafari) 

  

INTERVAL VALUED PENTAPARTITIONED NEUTROSOPHIC 
GRAPHS WITH AN APPLICATION TO MCDM 

Said Broumi 1,2*, D. Ajay 3, P. Chellamani 3, Lathamaheswari Malayalan 4, 
Mohamed Talea 1, Assia Bakali 5, Philippe Schweizer 6, Saeid Jafari 7 

1 Laboratory of Information Processing, Faculty of Science Ben M’Sik, University of 
Hassan II, Casablanca, Morocco  

2 Regional Center for the Professions of Education and Training (C.R.M.E.F), 
Casablanca-Settat, Morocco.  

3 Department of Mathematics, Sacred Heart College (Autonomous), Tamilnadu, India 
4 Department of Mathematics, Hindustan Institute of Technology & Science, Chennai, 

India 
5 Ecole Royale Navale-Boulevard Sour Jdid, Morocco 

6 Independent researcher, Switzerland 
7 College of Vestsjaelland South Herrestarede 11, Denmark 

 

Received: 20 June 2022 
Accepted: 27 August 2022 
First online: 03 October 2022 

Original scientific paper  

Abstract: The concept of interval valued pentapartitioned neutrosophic set is the 
extension of interval-valued neutrosophic set, quadripartitioned neutrosophic set, 
interval valued quadripartitioned neutrosophic set and pentapartitioned neutrosophic 
set. The powerful mathematical tool known as the interval valued pentapartitioned 
neutrosophic set divides indeterminacy into three separate components: unknown, 
contradiction, and ignorance. There are several applications for graph theory in 
everyday life, and it is a rapidly growing topic. The concept of an interval valued 
pentapartitioned neutrosophic set is used in graph theory. A decision-making method  
multicriteria (MCDM) is proposed by using the developed Interval valued 
Pentapartitioned Neutrosophic set with a numerical illustration. In this paper, as an 
extension of interval valued neutrosophic graph theory, we introduce the notions of 
Interval-Valued Pentapartitioned Neutrosophic Graph (IVPPN-graph) with degree, size, 
and order of an IVPPN-graph. 

Key words: Neutrosophic Set, Interval valued Pentapartitioned Neutrosophic sets, 
Neutrosophic Graph, IVPPN-Graph. 
  

mailto:author.%20broumisaid78@gmail.com
mailto:author.%20broumisaid78@gmail.com


 Interval Valued Pentapartitioned Neutrosophic Graphs with an Application to MCDM 
 

69 
 

1. Introduction 

Linguistic variables with uncertainties such as vagueness, imprecision and 
ambiguity are handled with a special type of mathematical tool known as fuzzy set 
theory (Zadeh, 1965). The contribution of research articles in the field of fuzzy sets in 
mathematics and wide range of applications are growing exponentially. The advantage 
of introducing Zadeh’s fuzzy sets in the place of classical sets has been its accuracy and 
precision in theory and compatibility and efficiency in case of applications. 

Fuzzy set has been extended to Intuitionistic fuzzy set (Atanasov, 1986). in which 
the membership and non-membership degree of the element of that set ranges 
between 0 and 1. Intuitionistic set was further extended to Interval valued fuzzy sets, 
Neutrosophic sets, Pythagorean sets and so on.  (Smarandache,1999 &2020) 
presented a new type of set as an extension of intuitionistic fuzzy set which is called 
Neutrosophic set. A neutrosophic set is characterized by a truth membership degree 
(T), an indeterminacy membership degree (I) and a falsity membership degree (F) 
independently, which are within the real unit interval ]0-,1+[ satisfying the condition 
that the total of the membership grades is within the range of 0 to 3. In (Wang, 2010) 
single valued neutrosophic set (SVNS) with set-theoretic operators was investigated 
by Wang et al. SVNSs have been developed into many new concepts and are applied in 
different disciplines (Broumi et al., 2016 & 2016a & 2016b &2016c; Chatterjee et al., 
2016&2016b). Pythagorean set (Yager, 2013). is an extension of the intutionistic set 
in which the condition differs from the intutionistic set that the sum of the squares the 
membership is less than 1.  

By combining the idea of fuzzy graphs and Pythagorean neutrosophic set Ajay and 
Chellamani et al. presented Pythagorean neutrosophic graphs (Ajay & Chellamani, 
2020) is a combination of the Pythagorean and Neutrosophic set in which the the sum 
of the squares of the membership, non-membership and indeterminacy membership 
lies in [0,2] and further theoretical concepts were developed in (Ajay & Chellamani, , 
2021 ; Chellamani & Ajay, 2021; Ajay et al., 2021 , 2022 ,2020a) ; Chellamani et 
al.,2021). The partition of indeterminacy function of the neutrosophic set into 
contradiction part and ignorance part is defined as the quadripartitioned single-
valued neutrosophic set (Chatterjee et al., 2016, 2020) 

(Quek et al, 2022) introduced the notion of pentapartitionned neutrosophic graphs 
(PPNGs), as an extended version of Single valued neutrosophic graphs. (Mallick & 
Pramanik, 2020) proposed the concept of interval-valued pentapartitionned 
neutrosophic sets (IVPPNSs) as a generalization of pentapartitionned neutrosophic 
sets in the spirit of interval-valued neutrosophic sets (Broumi, 2016) The concept of  
Interval valued pentapartitioned neutrosophic sets (IVPPNSs) allows a decision-
making expert to represented the membership degree contradiction membership 
function 𝐶𝐴(𝑥), an unknown membership function 𝑈𝐴(𝑥)  and a falsity membership 
degree of a set of options in terms of the interval; hence, the range of uncertain 
information they can describe is widened. Interval-valued pentapartitioned 
neutrosophic numbers (IVPPNNs) are the generalized version of pentapartitioned 
neutrosophic numbers, quadripartitioned neutrosophic numbers, interval-valued 
quadripartitioned neutrosophic numbers. IVPPNNs will help model problems with 
incomplete, uncertain and indeterminate information. And further theoretical 
concepts were developed in (Das et al, 2022& 2022a; Pramanik, 2022). (Hussain et al, 
2022) introduced the notion of quadripartitioned single-valued neutrosophic graphs 



S. Broumi et al./Oper. Res. Eng. Sci. Theor. Appl. 5(3)2022 68-91 
  

70 
 

and developed some operations on it. The Cartesian product, cross product, 
lexicographic product, strong product and composition of quadripartitioned single-
valued neutrosophic graph have been investigated. The proposed concepts are 
illustrated with an application in the climatic analysis of apple cultivation. (Kaufmann, 
1973), based on fuzzy relation (Zadeh, 2020).  developed the idea of fuzzy graphs 
(FGs). Later, (Rosenfeld, 1975) defined the basic properties of fuzzy relations which 
are generalized with fuzzy set as a base set and fuzzy analogues of graphic theoretical 
concepts like bridges and trees were established with their properties and further 
fuzzy graph concepts were developed and applied  in the real life situations in 
(Mohamed et al., 2020; Mordeson & Chang-Shyh, 1994; Naz et al, 2017 &2018; Quek, 
et al., 2018 ; Smarandache, 2013 ; Tan et al., 2021 ; Das et al., 2021 ; Majumder et al, 
2021; Saha et al., 2022). Recent developments of the fuzzy extensions are developed 
as in (Kumar et al., 2019 ; Radha et al.,2021 ; Vellapandi & Gunasekaran, 2020 ;Das & 
Edalatpanah, 2020 ; Polymenis, 2021; Mao et al, 2020 ; Voskoglou & Broumi, 2022 ; 
Zhang et al., 2022 ; Al-Hamido, 2022). To the best of authors knowledge, a very less 
work is being done on the interval-valued pentapartitioned neutrosophic set 
(IVPPNS), so the present study define the operations on the IVPPNS and later on 
extend it to graphs environment. The major contributions in this work are explained 
as follows: 

1. The notions of interval valued pentapartitioned neutrosophic Graphs (IVPPNGs) 
are introduced. This manuscript makes the first attempt in the literature about the 
concept in neutrosophic graphs. 

2. In addition, the complete and strong IVPPNG are defined. The operations like a 
Cartesian product, cross product, lexicographic product, strong product and the 
composition of IVPPNGs with their properties are discussed.  

3. In addition, the complete, strong and complement of IVPPNGs are defined. 

4. We extend some of the basic properties for this PNFG along with few examples. 
We have developed a decision-making model using the introduced Interval valued 
Pentapartitioned Neutrosophic graphs and applied it for a numerical illustration. A list 
of contribution (Table 1) of authors is presented below. 

Table 1. Contribution of authors to extension of neutrosophic graphs 
Authors Year Contributions 

Akram and Siddique 2017 Introduced neutrosophic competition graph  
Broumi et al. 2018 Introduced Generalized neutrosophic graph  

Das et al 2020 Introduced generalized neutrosophic 
competition graph  

Ajay and Chellamani  2020 Pythagorean Neutrosophic Fuzzy Graphs  
S. Satham Hussain 2022 Quadripartitioned Neutrosophic Soft Graphs  
S. Satham Hussain 2022 Quadripartitioned Bipolar Single Valued 

Neutrosophic Graph  
S. Satham Hussain 2022 Quadripartitioned Single-Valued Neutrosophic 

Graph  
Quek et al. 2022 Pentapartitioned Single-Valued Neutrosophic 

Graph  
Our approch 2022 Interval valued Pentapartitioned Single-Valued 

Neutrosophic Graph 



 Interval Valued Pentapartitioned Neutrosophic Graphs with an Application to MCDM 
 

71 
 

2. Preliminaries  

This section summarizes some basic concepts from the theory of IVNSs and the 
concept of IVPPNSs, which is the foundation for the concept of IVPPNGs. Further 
details on the NS, SVNS and IVNS theories can be found in [3-6] Smarandache (1999) 
and (Wang et al. 2010), respectively. 

Definition 2.1: [5] Let A and B be two SVNSs over a universe Y. 

(i) A is contained in B, if 𝑇𝐴(𝑦)  ≤ 𝑇𝐵 (𝑦),  𝐼𝐴(𝑦)  ≥ 𝐼𝐵 (𝑦), and 𝐹𝐴(𝑦)  ≥ 𝐹𝐵 (𝑦), for all 𝑦 ∈ 
Y . This relationship is denoted as A ⊆ B  (1) 

(ii) A and B are said to be equal if A ⊆ B and B ⊆ A  (2) 

(iii) 𝐴𝑐  = (𝑦, (𝐹𝐴(𝑦) , 1 − 𝐼𝐴(𝑦), 𝑇𝐴(𝑦))), for all 𝑦 ∈ Y  (3) 

(iv) A ∪ B = (𝑦, (max (𝑇𝐴, 𝑇𝐵 ), min (𝐼𝐴 , 𝐼𝐵 ), min (𝐹𝐴, 𝐹𝐵 ))), for all 𝑦 ∈ Y. (4) 

(v) A ∩ B = (𝑦, (min (𝑇𝐴, 𝑇𝐵 ), max (𝐼𝐴 , 𝐼𝐵 ), max (𝐹𝐴, 𝐹𝐵 ))), for all 𝑦 ∈ Y.  (5) 

Definition 2.2: (Wang et al, 2010) An Interval valued neutrosophic Sets (IVNS) A 
in X is denoted by an interval truth-membership function 𝑇𝐴(𝑥), an interval 
indterminacy-membership function 𝐼𝐴(𝑥),and an interval falsity membership function 
𝐹𝐴(𝑥)  for each point x in X, there are 

�̃�𝐴(𝑥) = [𝑇𝐴
𝐿 (𝑥), 𝑇𝐴

𝑈 (𝑥)] ⊆ [0,1], (6) 

𝐼𝐴(𝑥) = [𝐼𝐴
𝐿 (𝑥), 𝐼𝐴

𝑈 (𝑥)] ⊆ [0,1], (7) 

�̃�𝐴(𝑥) = [𝐹𝐴
𝐿 (𝑥), 𝐹𝐴

𝑈 (𝑥)] ⊆ [0,1]. (8) 

Therefore an IVNS A can be denoted as, A= {〈𝑥, �̃�𝐴(𝑥), 𝐼𝐴 (𝑥), , �̃�𝐴(𝑥)〉| 𝑥 ∈ 𝑋} 

Then the sum of �̃�𝐴(𝑥), 𝐼𝐴(𝑥), �̃�𝐴(𝑥) satisfies the condition  

0 ≤ 𝑇𝐴
𝑈 (𝑥) + 𝐼𝐴

𝑈 (𝑥) + 𝐹𝐴
𝑈 (𝑥) ≤ 3 (9) 

If the upper and lower ends of the interval values of 𝑇𝐴(𝑥), 𝐼𝐴 (𝑥) 𝑎𝑛𝑑 𝐹𝐴(𝑥) in an 
IVNS are equal then IVNS reduces to the SVNS.  

Definition 2.3: (Wang et al, 2010)  Let A and B be two IVNSs over a universe Y . 

(i) A is contained in B, if 𝑇𝐴
𝐿 (𝑦)  ≤ 𝑇𝐵

𝐿 (𝑦), 𝑇𝐴
𝑈 (𝑦)  ≤ 𝑇𝐵

𝑈 (𝑦) , 𝐼𝐴
𝐿 (𝑦)  ≥ 𝐼𝐵

𝐿 (𝑦), 𝐼𝐴
𝑈 (𝑦)  ≥ 𝐼𝐵

𝑈 (𝑦), 
and 𝐹𝐴

𝐿 (𝑦)  ≥ 𝐹𝐵
𝐿 (𝑦), 𝐹𝐴

𝑈 (𝑦)  ≥ 𝐹𝐵
𝑈 (𝑦), for all 𝑦 ∈ Y. This relationship is denoted as  

A ⊆ B. (10) 

(ii) A and B are said to be equal if A ⊆ B and B ⊆ A.  (11) 

(iii) 𝐴𝑐  = (𝑦, (𝐹𝐴(𝑦) , 1 − 𝐼𝐴(𝑦), 𝑇𝐴(𝑦))), for all 𝑦 ∈ Y.  (12) 

(iv) A ∪ B = (𝑦, (max(𝑇𝐴, 𝑇𝐵 ), min(𝐼𝐴, 𝐼𝐵 ), min(𝐹𝐴, 𝐹𝐵 ))), for all 𝑦 ∈ Y.  (13) 

(v) A ∩ B = (𝑦, (min(𝑇𝐴, 𝑇𝐵 ), max(𝐼𝐴, 𝐼𝐵 ), max(𝐹𝐴, 𝐹𝐵 ))), for all 𝑦 ∈ Y  (14) 

Definition 2.4: (Chatterjee eta al., 2016) Quadripartitioned neutrosophic sets 

Let X be a universe. A Quadripartitioned neutrosophic set A with independent 
neutrosophic components on X is an object of the form  

𝐴 =  {< 𝑥, 𝑇𝐴(𝑥), 𝐶𝐴(𝑥), 𝑈𝐴(𝑥), 𝐹𝐴(𝑥) >: 𝑥 ∈ X} (15) 



S. Broumi et al./Oper. Res. Eng. Sci. Theor. Appl. 5(3)2022 68-91 
  

72 
 

𝑎𝑛𝑑 0 ≤ 𝑇𝐴 (𝑥) + 𝐶𝐴(𝑥) + 𝑈𝐴(𝑥) + 𝐹𝐴 (𝑥) ≤ 4 (16) 

Here, 𝑇𝐴(𝑥) is the truth membership,  𝐶𝐴(𝑥) is contradiction membership, 𝑈𝐴 (𝑥) is 
ignorance membership and 𝐹𝐴(𝑥) is the false membership. 

Definition 2.5: (Pramanik, 2022). Interval quadripartionned neutrosophic 
Sets 

An Interval quadripartionned neutrosophic Sets (IQNS) A in X is denoted by truth-
membership function 𝑇𝐴(𝑥), a contradiction membership function 𝐶𝐴(𝑥),  an unknown 
membership function 𝑈𝐴(𝑥)  and a falsity membership function 𝐹𝐴(𝑥), for each point x 
in X, there are 

𝑇𝐴 (𝑥) = [𝑇𝐴
𝐿 (𝑥), 𝑇𝐴

𝑈 (𝑥)] ⊆ [0,1], 𝐶𝐴(𝑥) = [𝐶𝐴
𝐿 (𝑥), 𝑇𝐴

𝑈 (𝑥)] ⊆ [0,1],   

𝑈𝐴(𝑥) = [𝑈𝐴
𝐿 (𝑥), 𝑈𝐴

𝑈 (𝑥)] ⊆ [0,1], 𝐹𝐴(𝑥) = [𝐹𝐴
𝐿 (𝑥), 𝐹𝐴

𝑈 (𝑥)] ⊆ [0,1],  

Therefore an IQNS A can be denoted as, 

A= {〈𝑥, 𝑇𝐴 (𝑥), 𝐶𝐴(𝑥), 𝑈𝐴(𝑥), 𝐹𝐴(𝑥)〉| 𝑥 ∈ 𝑋} (17) 

A={〈
𝑥,[𝑇𝐴

𝐿(𝑥),𝑇𝐴
𝑈

(𝑥)],[𝐶𝐴
𝐿(𝑥),𝑇𝐴

𝑈
(𝑥)]

[𝑈𝐴
𝐿 (𝑥),𝑈𝐴

𝑈(𝑥)],[𝐹𝐴
𝐿(𝑥),𝐹𝐴

𝑈(𝑥)]
〉 | 𝑥 ∈ 𝑋}  (18) 

Then the sum of 𝑇𝐴(𝑥), 𝐶𝐴(𝑥), 𝑈𝐴(𝑥), 𝐹𝐴(𝑥) satisfies the condition  

0 ≤ 𝑇𝐴
𝑈 (𝑥) + 𝐶𝐴

𝑈 (𝑥) + 𝑈𝐴
𝑈 (𝑥) + 𝐹𝐴

𝑈 (𝑥) ≤ 4  (19) 

If the upper and lower bounds of the interval values of 
𝑇𝐴(𝑥), 𝐶𝐴(𝑥), 𝑈𝐴(𝑥) 𝑎𝑛𝑑 𝐹𝐴(𝑥) in an IQNS are equal then IQNS reduces to the QSVNS.  

Definition 2.6: (Mallick & Pramanik, 2020). Pentapartitioned neutrosophic sets 

Let X be a universe. A Pentapartitioned neutrosophic set A with independent 
neutrosophic components on X is an object of the form  

𝐴 =  {< 𝑥, 𝑇𝐴(𝑥), 𝐶𝐴(𝑥), 𝑈𝐴(𝑥), 𝐺𝐴(𝑥), 𝐹𝐴(𝑥) >: 𝑥 ∈ X}  (20) 

𝑎𝑛𝑑 0 ≤ 𝑇𝐴 (𝑥) + 𝐶𝐴(𝑥) + 𝑈𝐴(𝑥) + 𝐺𝐴 (𝑥) + 𝐹𝐴 (𝑥) ≤ 5 (21) 

Here, 𝑇𝐴(𝑥) is the truth membership,  𝐶𝐴(𝑥) is contradiction membership, 𝑈𝐴 (𝑥) is 
ignorance membership, 𝐺𝐴(𝑥) 𝑖𝑠 ignorance membership, and 𝐹𝐴(𝑥) is the false 
membership. 

Definition 2.7: (Das et al., 2022) Interval Pentapartionned neutrosophic sets 

An Interval Pentapartionned neutrosophic Sets (IPNS) A in X is denoted by truth-
membership function 𝑇𝐴(𝑥), a contradiction membership function 𝐶𝐴(𝑥),  an unknown 
membership function 𝑈𝐴(𝑥), 𝐺𝐴(𝑥)𝑖𝑠 ignorance membership and a falsity membership 
function 𝐹𝐴(𝑥) for each point x in X, there are 

𝑇𝐴 (𝑥) = [𝑇𝐴
𝐿 (𝑥), 𝑇𝐴

𝑈 (𝑥)] ⊆ [0,1], 𝐶𝐴(𝑥) = [𝐶𝐴
𝐿 (𝑥), 𝑇𝐴

𝑈 (𝑥)] ⊆ [0,1],  

𝑈𝐴(𝑥) = [𝑈𝐴
𝐿 (𝑥), 𝑈𝐴

𝑈 (𝑥)] ⊆ [0,1], 𝐺𝐴(𝑥) = [𝐺𝐴
𝐿 (𝑥), 𝐺𝐴

𝑈 (𝑥)] ⊆ [0,1],  

 𝐹𝐴(𝑥) = [𝐹𝐴
𝐿 (𝑥), 𝐹𝐴

𝑈 (𝑥)] ⊆ [0,1],  

Therefore an IPNS A can be denoted as, 

A= {〈𝑥, 𝑇𝐴 (𝑥), 𝐶𝐴(𝑥), 𝑈𝐴(𝑥), 𝐺𝐴(𝑥), 𝐹𝐴(𝑥)〉| 𝑥 ∈ 𝑋}  (22) 



 Interval Valued Pentapartitioned Neutrosophic Graphs with an Application to MCDM 
 

73 
 

A={〈
𝑥,[𝑇𝐴

𝐿(𝑥),𝑇𝐴
𝑈

(𝑥)],[𝐶𝐴
𝐿(𝑥),𝑇𝐴

𝑈
(𝑥)]

[𝑈𝐴
𝐿 (𝑥),𝑈𝐴

𝑈(𝑥)],[𝐺𝐴
𝐿(𝑥),𝐺𝐴

𝑈(𝑥)],[𝐹𝐴
𝐿(𝑥),𝐹𝐴

𝑈(𝑥)]
〉 | 𝑥 ∈ 𝑋} 

Then the sum of 𝑇𝐴(𝑥), 𝐶𝐴(𝑥), 𝑈𝐴(𝑥), 𝐺𝐴(𝑥), 𝐹𝐴(𝑥) satisfies the condition  

0 ≤ 𝑇𝐴
𝑈 (𝑥) + 𝐶𝐴

𝑈 (𝑥) + 𝑈𝐴
𝑈 (𝑥) + 𝐺𝐴

𝑈 (𝑥) + 𝐹𝐴
𝑈 (𝑥) ≤ 5 (23) 

If the upper and lower bounds of the interval values of 
𝑇𝐴(𝑥), 𝐶𝐴(𝑥), 𝑈𝐴(𝑥), 𝐺𝐴(𝑥)  𝑎𝑛𝑑 𝐹𝐴(𝑥) in an IPNS are equal then IPNS reduces to the 
PSVNS. 

Definition 2.8: (Broumi et al., 2016) Let V be a set. Let E ⊆ {{𝜇, 𝑣}: 𝜇, 𝑣 ∈

𝑉 𝑤𝑖𝑡ℎ 𝜇 ≠  𝑣}. 

Let A be an IVNS on V and B be an IVNS on E with  

𝑇𝐵
𝑙 (𝜇, 𝑣) ≤ min{𝑇𝐴

𝑙 (𝜇), 𝑇𝐴
𝑙 (𝑣)}, 𝑇𝐵

𝑢(𝜇, 𝑣)  ≤ min{𝑇𝐴
𝑢 (𝜇), 𝑇𝐴

𝑢 (𝑣)} (24) 

𝐼𝐵
𝑙 (𝜇, 𝑣) ≥ max{𝐼𝐴

𝑙 (𝜇), 𝐼𝐴
𝑙 (𝑣)}, 𝐼𝐵

𝑢(𝜇, 𝑣)  ≥ max{𝐼𝐴
𝑢(𝜇), 𝐼𝐴

𝑢(𝑣)} (25) 

𝐹𝐵
𝑙 (𝜇, 𝑣)  ≥ min{𝐹𝐴

𝑙 (𝜇), 𝐹𝐴
𝑙 (𝑣)}, 𝐹𝐵

𝑢(𝜇, 𝑣)  ≥ min{𝐹𝐴
𝑢(𝜇), 𝐹𝐴

𝑢(𝑣)}for all {𝜇, 𝑣}∈ E. Then 
G=(A, B, V, E) is said to be an Interval-valued neutrosophic graph.  (26) 

Definition 2.9: [22] Let V be a set. Let 𝐸⊆ {{𝑣, 𝜔}: 𝑣, 𝜔 ∈ 𝑉 ′ 𝑤𝑖𝑡ℎ 𝜇 ≠  𝑣}. Let A be 

a PPNS on V, and B be a PPNS on E, with 𝑡𝐵 (𝜇, 𝑣) ≤  min{𝑡𝐴(𝜇), 𝑡𝐴(𝑣)}, 𝑐𝐵 (𝜇, 𝑣) ≥ 
max{𝑐𝐴(𝜇), 𝑐𝐴(𝑣)}, 𝑔𝐵 (𝜇, 𝑣) ≥ max{𝑔𝐴(𝜇), 𝑔𝐴(𝑣)}, 𝑢𝐵 (𝜇, 𝑣) ≥ max{𝑢𝐴(𝜇), 𝑢𝐴(𝑣)}and 
𝑓𝐵 (𝜇, 𝑣) ≥ max{𝑓𝐴(𝜇), 𝑓𝐴(𝑣)}for all { 𝜇, 𝑣 }∈ E. Then we have the following: 

(i) G=(A, B, V, E) is said to be a pentapartitioned neutrosophic graph (PPNG).  

(ii) Each 𝑣 ∈ V is said to be a vertex of G.            

(iii) Each{𝜇, 𝑣}∈ E is said to be an edge of G. 

3. Interval Pentapartitioned neutrosophic graphs 

In this section, first the concept of interval pentapartitioned neutrosophic graphs 
(IVPPNGs) is introduced. Then based on the definitions of IVNS, interval-valued 
neutrosophic graphs [24] and IVPPNS given in definition 2.6 and definition 2.7, 
respectively will be used to put forward the novel concept of IVPPNGs. The related 
properties pertaining to this concept will be subsequently investigated. 

Definition 3.1. Let V be a set. Let 𝐸⊆ {{𝜇, 𝑣}: 𝑣, 𝜇 ∈ 𝑉 𝑤𝑖𝑡ℎ 𝜇 ≠  𝑣}. Let A be an 

IVPPNS on V, and B be an IVPPNS on E, with  

𝑡𝐵
𝑙 (𝜇, 𝑣) ≤  min{𝑡𝐴

𝑙 (𝜇), 𝑡𝐴
𝑙 (𝑣)}, 𝑡𝐵

𝑢(𝜇, 𝑣) ≤  min{𝑡𝐴
𝑢(𝜇), 𝑡𝐴

𝑢(𝑣)}, (27) 

𝑐𝐵
𝑙 (𝜇, 𝑣) ≥ max{𝑐𝐴

𝑙 (𝜇), 𝑐𝐴
𝑙 (𝑣)},𝑐𝐵

𝑢 (𝜇, 𝑣) ≥ max{𝑐𝐴
𝑢 (𝜇), 𝑐𝐴

𝑢 (𝑣)},   (28) 

𝑔𝐵
𝑙 (𝜇, 𝑣) ≥ max{𝑔𝐴

𝑙 (𝜇), 𝑔𝐴
𝑙 (𝑣)},𝑔𝐵

𝑢(𝜇, 𝑣) ≥ max{𝑔𝐴
𝑢 (𝜇), 𝑔𝐴

𝑢 (𝑣)}, (29) 

𝑢𝐵
𝑙 (𝜇, 𝑣) ≥ max{𝑢𝐴

𝑙 (𝜇), 𝑢𝐴
𝑙 (𝑣)},𝑢𝐵

𝑢(𝜇, 𝑣) ≥ max{𝑢𝐴
𝑢(𝜇), 𝑢𝐴

𝑢(𝑣)},  (30) 

and 𝑓𝐵
𝑙 (𝜇, 𝑣) ≥ max{𝑓𝐴

𝑙 (𝜇), 𝑓𝐴
𝑙 (𝑣)},𝑓𝐵

𝑢 (𝜇, 𝑣) ≥ max{𝑓𝐴
𝑢(𝜇), 𝑓𝐴

𝑢(𝑣)}, (31) 

for all { 𝜇, 𝑣 }∈ E. Then we have the following: 



S. Broumi et al./Oper. Res. Eng. Sci. Theor. Appl. 5(3)2022 68-91 
  

74 
 

(i) G=(A, B, V, E) is said to be an interval pentapartitioned neutrosophic graph 
(IVPPNG). (32) 

(ii) Each 𝑣 ∈ V is said to be a vertex of G. (33) 

(iii) Each{ 𝜇, 𝑣 }∈ E is said to be an edge of G. (34) 

Fig. 1 A graphical representation of the PPNG G. 

Notation 3.1.1 Let G = (A, B, V, E) be a IVPPNG. Denote 𝑚𝐴 : V → [0, 1]
5, where 

𝑚𝐴(𝑣)=(�̃�𝐴(𝑣), �̃�𝐴(𝑣), �̃�𝐴(𝑣), �̃�𝐴(𝑣), 𝑓𝐴(𝑣)) for all 𝑣 ∈ V. Denote 𝑚𝐵 : E →Int( [0, 1]
5), 

where 𝑚𝐵 (𝜇, 𝑣 ) =(�̃�𝐵 (𝜇, 𝑣), �̃�𝐵 (𝜇, 𝑣), �̃�𝐵 (𝜇, 𝑣), �̃�𝐵 (𝜇, 𝑣), 𝑓𝐵 (𝜇, 𝑣)) for all { 𝜇, 𝑣 } ∈ E. 

Example 3.2 Let G = (A, B, V, E) be an IVPPNG with V = {𝑣1, 𝑣2, 𝑣3, 𝑣4, 𝑣5} and E = 
{{𝑣1, 𝑣2}, {𝑣1, 𝑣3}, {v1,v4}, {𝑣2, 𝑣5}, {𝑣4, 𝑣5}}. Then we have the following (Fig. 1): 

 

Figure 1. Interval Valued Pentapartitioned Neutrosophic Graph 

𝑚𝐴(𝑣1) = ([. 7, .9], [. 6, .7], [. 5, .7], [. 4, .6], [. 3, .5]), 

𝑚𝐴(𝑣2) = ([. 6, .7], [. 5, .8], [. 3, .6], [. 4, .9], [. 1, .5]), 

𝑚𝐴(𝑣3) = ([. 4, .8], [. 5, .7], [. 6, .9], [. 7, .9], [. 2, .5]), 

𝑚𝐴(𝑣4) = ([. 3, .6], [. 4, .7], [. 5, .9], [. 7, .9], [. 2, .5]), 

𝑚𝐵 (𝑣5) = ([. 7, .9], [. 6, .8], [. 5, .7], [. 3, .6], [. 2, .5]). 

𝑚𝐵 (𝑣1𝑣2) = ([. 6, .7], [. 7, .9], [. 5, .7], [. 5, .9], [. 4, .6]), 

𝑚𝐵 (𝑣1𝑣3) = ([. 4, .8], [. 6, .8], [. 6, .9], [. 4, .6], [. 3, .6]), 

𝑚𝐵 (𝑣1𝑣4) = ([. 3, .6], [. 6, .8], [. 5, .9], [. 7, .9], [. 3, .5]), 

𝑚𝐵 (𝑣2𝑣5) = ([. 6, .7], [. 6, .8], [. 5, .7], [. 4, .9], [. 2, .6]), 

𝑚𝐵 (𝑣4𝑣5) = ([. 3, .6], [. 6, .8], [. 5, .9], [. 7, .9], [. 3, .7]). 

Definition 3.3. Let G = (A, B, V, E) and H = (𝐴′, 𝐵′, 𝑉 ′, 𝐸′)  be two IVPPNGs that 
satisfies the following conditions: 

(i) 𝑉 ′⊆ V (35) 



 Interval Valued Pentapartitioned Neutrosophic Graphs with an Application to MCDM 
 

75 
 

(ii) 𝐸′⊆ {{𝑣, 𝜇}: 𝑣, 𝜇 ∈ 𝑉 ′ 𝑤𝑖𝑡ℎ 𝜇 ≠  𝑣}  (36) 

(iii) 𝑚𝐴′ (𝑣)= 𝑚𝐴(𝑣), for all 𝑣 ∈ 𝑉
′⊆ V (37) 

(iv) 𝑚𝐵′ (𝑣, 𝜇)=𝑚𝐵 (𝑣, 𝜇), for all {𝑣, 𝜇} ∈ 𝐸
′ ⊆ E. (38) 

Then, H is said to be a partial interval valued pentapartitioned neutrosophic graph 
(partial-IVPPNG) of G. 

Definition 3.4. Let G = (A, B, V, E) and H = (𝐴′, 𝐵′, 𝑉 ′, 𝐸′)  be two IVPPNGs that 
satisfies the following conditions: 

(i) 𝑉 ′⊆ V  (39) 

(ii) 𝐸′⊆ {{𝑣, 𝜔}: 𝑣, 𝜔 ∈ 𝑉 ′ 𝑤𝑖𝑡ℎ 𝜔 ≠  𝑣} (40) 

(iii) 𝑡
𝐴′
𝑙 (𝑣) ≤ 𝑡𝐴

𝑙 (𝑣), 𝑡
𝐴′
𝑢 (𝑣) ≤ 𝑡𝐴

𝑢(𝑣), 𝑐
𝐴′
𝑙 (𝑣) ≥ 𝑐𝐴

𝑙 (𝑣), 𝑐
𝐴′
𝑢 (𝑣) ≥ 𝑐𝐴

𝑢 (𝑣), 𝑔
𝐴′
𝑙 (𝑣) ≥ 𝑔𝐴

𝑙 (𝑣), 𝑔
𝐴′
𝑢 (𝑣) 

≥ 𝑔𝐴
𝑢 (𝑣), 𝑢

𝐴′
𝑙 (𝑣) ≥ 𝑢𝐴

𝑙 (𝑣), 𝑢
𝐴′
𝑢 (𝑣) ≥ 𝑢𝐴

𝑢(𝑣), 𝑓
𝐴′
𝑙 (𝑣) ≥ 𝑓𝐴

𝑙(𝑣), 𝑓
𝐴′
𝑢 (𝑣) ≥ 𝑓𝐴

𝑢(𝑣) for all 𝑣 ∈ 𝑉 ′⊆V

 (41) 

(iv) 𝑡
𝐵′
𝑙 (𝑣, 𝜔) ≤ 𝑡𝐵

𝑙 (𝑣, 𝜔), 𝑡
𝐵′
𝑢 (𝑣, 𝜔) ≤ 𝑡𝐵

𝑢(𝑣, 𝜔), 𝑐
𝐵′
𝑙 (𝑣, 𝜔) ≥ 𝑐𝐵

𝑙 (𝑣, 𝜔), 𝑐
𝐵′
𝑢 (𝑣, 𝜔) ≥ 𝑐𝐵

𝑢 (𝑣, 𝜔), 

𝑔
𝐵′
𝑙 (𝑣, 𝜔) ≥ 𝑔𝐵

𝑙 (𝑣), 𝑔
𝐵′
𝑢 (𝑣, 𝜔) ≥ 𝑔𝐵

𝑢(𝑣, 𝜔), 𝑢
𝐵′
𝑙 (𝑣, 𝜔) ≥ 𝑢𝐵

𝑙 (𝑣, 𝜔), 𝑢
𝐵′
𝑢 (𝑣, 𝜔) ≥ 𝑢𝐵

𝑢 (𝑣, 𝜔), 

𝑓
𝐵′
𝑙 (𝑣, 𝜔) ≥ 𝑓𝐵

𝑙 (𝑣, 𝜔), 𝑓
𝐵′
𝑢 (𝑣, 𝜔) ≥ 𝑓𝐵

𝑢(𝑣, 𝜔), for all { 𝑣, 𝜔 } ∈ 𝐸′ ⊆ E. (42) 

Then, H is said to be an interval valued pentapartitioned neutrosophic subgraph 
(IVPPNSG) of G. 

Example 3.5. Let 𝐺1be an IVPPNG and 𝐻1, 𝐻2 be the partial-IVPPNG and IVPPNSG 
of 𝐺1, respectively. The graphical representation of 𝐺1, 𝐻1 and 𝐻2 are shown in Figs. 2, 
3 and 4, respectively. Here 𝐻2 is a IVPPNSG of 𝐺1 but not a partial-IVPPNG of 𝐺1. 

𝑚𝐴(𝑣1) = ([. 7, .9], [. 6, .7], [. 5, .7], [. 4, .6], [. 3, .5]), 

𝑚𝐴(𝑣2) = ([. 6, .7], [. 5, .8], [. 3, .6], [. 4, .9], [. 1, .5]), 

𝑚𝐴(𝑣3) = ([. 4, .8], [. 5, .7], [. 6, .9], [. 7, .9], [. 2, .5]), 

𝑚𝐴(𝑣4) = ([. 3, .6], [. 4, .7], [. 5, .9], [. 7, .9], [. 2, .5]), 

𝑚𝐵 (𝑣5) = ([. 7, .9], [. 6, .8], [. 5, .7], [. 3, .6], [. 2, .5]). 

 

Figure 2. Interval Valued Pentapartitioned Neutrosophic Graph 



S. Broumi et al./Oper. Res. Eng. Sci. Theor. Appl. 5(3)2022 68-91 
  

76 
 

 

Figure 3. Partial Interval Valued Pentapartitioned Neutrosophic Graph 

 

Figure 4. Interval Valued Pentapartitioned Neutrosophic Subgraph 

Definition 3.6. Let G = (A, B, V, E) be an IVPPNG. Let 𝑣0 , 𝑣1,..., 𝑣𝑛 ∈ V , with {𝑣𝑖 , 𝑣𝑖+1} 
∈ E for all 0 < i ≤ n-1, and with 𝑣𝑖 ≠ 𝑣𝑗  for all 0 ≤ i < j ≤ n-1. Then we have the following: 

P =(𝑣0, 𝑣1,..., 𝑣𝑛) is said to be a an interval valued pentapartitioned neutrosophic 
path (IVPPNP) in G. 

For each i, {𝑣𝑖 , 𝑣𝑖+1} is said to be an edge of P. 

n is said to be the length of P. 



 Interval Valued Pentapartitioned Neutrosophic Graphs with an Application to MCDM 
 

77 
 

Definition 3.7. Let G = (A, B, V, E) be an IVPPNG. Then G is said to be connected if 
there exists at least one { 𝑣, 𝜔 } ∈ E for all 𝑣 ∈ V. 

Definition 3.8. Let G =(A, B, V, E) be an IVPPNG. Then 𝑣 ∈ V is said to be isolated if 
{ 𝑣, 𝜔 } ∉ / E for all 𝜔 ∈ V \ { 𝑣 }. the IVPPNG has 𝑣1 as the isolated vertex. 

Definition 3.9. Let G = (A, B, V, E) be an IVPPNG and let 𝑣 ∈ V. The degree of v, 
written as d(v), is defined as d(v) =∑ 𝑚𝐵𝜇∈𝑉 𝑤𝑖𝑡ℎ{𝑣,𝜔}∈𝐸 (𝑣, 𝜔). 

Remark 3.9.1. It follows that d(𝑣) ∈ [0, 1]5. 

Example 3.10. In Fig. 3 under Eg. 3.5, the degree of the vertices are as follows. 

d(𝑣1) = (0.5, 1.7, 1.5, 1.1, 1.2), d(𝑣3) =(0.6, 1.7, 1.7, 1.2, 1.2), d(𝑣4)= (0.5, 1.8, 1.6, 1.3, 
1.6). 

Definition 3.11. Let G =(A, B, V, E) be an IVPPNG and let P =(𝑣0, 𝑣1,..., 𝑣𝑛) be an 
IVPPNP in G. The strength of P, denoted as s(P), is defined as :   
s(P)=

 ([𝑠
𝑡𝐴

𝑙 (𝑃), 𝑠𝑡𝐴
𝑢 (𝑃)]  , [𝑠

𝑐𝐴
𝑙 (𝑃), 𝑠𝑐𝐴

𝑢 (𝑃)] , [𝑠
𝑔𝐴

𝑙 (𝑃), 𝑠𝑔𝐴
𝑢 (𝑃)]  , [𝑠

𝑢𝐴
𝑙 (𝑃), 𝑠𝑢𝐴

𝑢 (𝑃)] , [𝑠
𝑓𝐴

𝑙 (𝑃), 𝑠𝑓𝐴
𝑢 (𝑃)]) 

= (𝑠𝑡 (𝑃), 𝑠𝑐̃ (𝑃), 𝑠�̃� (𝑃), 𝑠𝑢 (𝑃), 𝑠�̃� (𝑃)) (43) 

where 

 𝑠
𝑡𝐴

𝑙 (𝑃)= 𝑚𝑖𝑛{𝑡𝐵
𝑙 (𝑣𝑖 , 𝑣𝑖+1): 0 ≤ 𝑖 ≤ 𝑛 − 1}, 𝑠𝑡𝐴

𝑢 (𝑃)= 𝑚𝑖𝑛{𝑡𝐵
𝑢(𝑣𝑖 , 𝑣𝑖+1) ∶  0 ≤ 𝑖 ≤ 𝑛 − 1}, 

𝑠
𝑐𝐴

𝑙 (𝑃)= 𝑚𝑎𝑥{𝑐𝐵
𝑙 (𝑣𝑖 , 𝑣𝑖+1) ∶  0 ≤ 𝑖 ≤ 𝑛 − 1}, 𝑠𝑐𝐴

𝑢 (𝑃)= 𝑚𝑎𝑥{𝑐𝐵
𝑢(𝑣𝑖 , 𝑣𝑖+1) ∶  0 ≤ 𝑖 ≤ 𝑛 −

1}, 

𝑠
𝑔𝐴

𝑙 (𝑃)= 𝑚𝑎𝑥{𝑔𝐵
𝑙 (𝑣𝑖 , 𝑣𝑖+1) ∶  0 ≤ 𝑖 ≤ 𝑛 − 1}, 𝑠𝑔𝐴

𝑢 (𝑃)= 𝑚𝑎𝑥{𝑔𝐵
𝑢 (𝑣𝑖 , 𝑣𝑖+1) ∶ 0 ≤ 𝑖 ≤ 𝑛 −

1}, 

𝑠
𝑢𝐴

𝑙 (𝑃)= 𝑚𝑎𝑥{𝑢𝐵
𝑙 (𝑣𝑖 , 𝑣𝑖+1) ∶  0 ≤ 𝑖 ≤ 𝑛 − 1}, 𝑠𝑢𝐴

𝑢 (𝑃)= 𝑚𝑎𝑥{𝑢𝐵
𝑢(𝑣𝑖 , 𝑣𝑖+1) ∶  0 ≤ 𝑖 ≤ 𝑛 −

1}, and 

𝑠
𝑓𝐴

𝑙 (𝑃)= 𝑚𝑎𝑥{𝑓𝐵
𝑙 (𝑣𝑖 , 𝑣𝑖+1) ∶  0 ≤ 𝑖 ≤ 𝑛 − 1}, 𝑠𝑓𝐴

𝑢 (𝑃)= 𝑚𝑎𝑥{𝑓𝐵
𝑢(𝑣𝑖 , 𝑣𝑖+1) ∶  0 ≤ 𝑖 ≤ 𝑛 −

1}, Moreover, the strength of connectedness among the vertices a, b ∈ V in G, denoted 

as 𝑟𝐺 (a, b), is defined as: 

𝑟𝐺 (a, b)=(𝑟𝑡,𝐺 (a, b), 𝑟𝑐̃,𝐺 (a, b), 𝑟�̃�,𝐺 (a, b), 𝑟𝑢,𝐺 (a, b), 𝑟�̃�,𝐺 (a, b))  With (44) 

𝑟𝑡,𝐺 (a, b) =   [𝑟𝑡𝐴
𝑙

,𝐺
(a, b), 𝑟𝑡𝐴

𝑢
,𝐺 (a, b)],    𝑟𝑐̃,𝐺 (a, b) =   [𝑟𝑐𝐴

𝑙
,𝐺

(a, b), 𝑟𝑐𝐴
𝑢

,𝐺 (a, b)]   

𝑟�̃�,𝐺 (a, b) =   [𝑟𝑔𝐴
𝑙

,𝐺
(a, b), 𝑟𝑔𝐴

𝑢
,𝐺 (a, b)],    𝑟𝑢,𝐺 (a, b) =   [𝑟𝑢𝐴

𝑙
,𝐺

(a, b), 𝑟𝑢𝐴
𝑢

,𝐺 (a, b)] and 

𝑟�̃�,𝐺 (a, b) =   [𝑟𝑓𝐴
𝑙

,𝐺
(a, b), 𝑟𝑓𝐴

𝑢
,𝐺 (a, b)]   

Where 

𝑟
𝑡𝐴

𝑙
,𝐺

(a, b) =  𝑚𝑎𝑥 {𝑠
𝑡𝐴

𝑙 (𝑃): P = (𝑣0, 𝑣1, . . . , 𝑣𝑛𝑃) in G 𝑤𝑖𝑡ℎ 𝑣0 = 𝑎 𝑎𝑛𝑑 𝑣𝑛𝑃 = 𝑏}, 

𝑟𝑡𝐴
𝑢

,𝐺 (a, b) =  𝑚𝑎𝑥{𝑠𝑡𝐴
𝑢 (𝑃): P = (𝑣0, 𝑣1, . . . , 𝑣𝑛𝑃 ) in G 𝑤𝑖𝑡ℎ 𝑣0 = 𝑎 𝑎𝑛𝑑 𝑣𝑛𝑃 = 𝑏}, 

𝑟
𝑐𝐴

𝑙
,𝐺

(a, b) =  𝑚𝑖𝑛 {𝑠
𝑐𝐴

𝑙 (𝑃): P = (𝑣0, 𝑣1, . . . , 𝑣𝑛𝑃) in G 𝑤𝑖𝑡ℎ 𝑣0 = 𝑎 𝑎𝑛𝑑 𝑣𝑛𝑃 = 𝑏}, 



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78 
 

𝑟𝑐𝐴
𝑢

,𝐺 (a, b) =  𝑚𝑖𝑛{𝑠𝑐𝐴
𝑢 (𝑃): P = (𝑣0, 𝑣1, . . . , 𝑣𝑛𝑃) in G 𝑤𝑖𝑡ℎ 𝑣0 = 𝑎 𝑎𝑛𝑑 𝑣𝑛𝑃 = 𝑏}, 

𝑟
𝑔𝐴

𝑙
,𝐺

(a, b) =  𝑚𝑖𝑛 {𝑠
𝑔𝐴

𝑙 (𝑃): P = (𝑣0, 𝑣1, . . . , 𝑣𝑛𝑃 ) in G 𝑤𝑖𝑡ℎ 𝑣0 = 𝑎 𝑎𝑛𝑑 𝑣𝑛𝑃 = 𝑏}, 

𝑟𝑔𝐴
𝑢

,𝐺 (a, b) =  𝑚𝑖𝑛{𝑠𝑔𝐴
𝑢 (𝑃) : P = (𝑣0, 𝑣1, . . . , 𝑣𝑛𝑃 ) in G 𝑤𝑖𝑡ℎ 𝑣0 = 𝑎 𝑎𝑛𝑑 𝑣𝑛𝑃 = 𝑏}, 

𝑟
𝑢𝐴

𝑙
,𝐺

(a, b) =  𝑚𝑖𝑛 {𝑠
𝑢𝐴

𝑙 (𝑃): P = (𝑣0, 𝑣1, . . . , 𝑣𝑛𝑃 ) in G 𝑤𝑖𝑡ℎ 𝑣0 = 𝑎 𝑎𝑛𝑑 𝑣𝑛𝑃 = 𝑏}, 

𝑟𝑢𝐴
𝑢

,𝐺 (a, b) =  𝑚𝑖𝑛{𝑠𝑢𝐴
𝑢 (𝑃): P = (𝑣0, 𝑣1, . . . , 𝑣𝑛𝑃 ) in G 𝑤𝑖𝑡ℎ 𝑣0 = 𝑎 𝑎𝑛𝑑 𝑣𝑛𝑃 = 𝑏}, 

𝑟
𝑓𝐴

𝑙
,𝐺

(a, b) =  𝑚𝑖𝑛 {𝑠
𝑓𝐴

𝑙 (𝑃): P = (𝑣0, 𝑣1, . . . , 𝑣𝑛𝑃 ) in G 𝑤𝑖𝑡ℎ 𝑣0 = 𝑎 𝑎𝑛𝑑 𝑣𝑛𝑃 = 𝑏}, 

𝑟𝑓𝐴
𝑢

,𝐺 (a, b) =  𝑚𝑖𝑛{𝑠𝑓𝐴
𝑢 (𝑃): P = (𝑣0, 𝑣1, . . . , 𝑣𝑛𝑃) in G 𝑤𝑖𝑡ℎ 𝑣0 = 𝑎 𝑎𝑛𝑑 𝑣𝑛𝑃 = 𝑏}. 

Definition 3.12. Let G = (A, B, V, E) be an IVPPNG and and let {𝑣, 𝜔} be an edge in 
G. Denote 𝐺{𝑣,𝜔} 

′  as the partial-IVPPNG of G, in which 𝐺{𝑣,𝜔} 
′ =(𝐴′, 𝐵′, 𝑉 ′, 𝐸′)  with 𝑉 = 

𝑉 ′and 𝐸′={{𝑣, 𝜔}}. Then, {𝑣, 𝜔} is said to be an interval valued pentapartitioned 
neutrosophic bridge (IVPPNB) in G if at least one of the following conditions holds for 
some a,b ∈ V. 

(i) 𝑟𝑡,𝐺{𝑣,𝜔} 
′ (a, b) < 𝑟𝑡,𝐺 (a, b) (45) 

(ii) 𝑟𝑐̃,𝐺{𝑣,𝜔} 
′ (a, b) > 𝑟𝑐̃,𝐺 (a, b) (46) 

(iii) 𝑟�̃�,𝐺{𝑣,𝜔} 
′ (a, b) > 𝑟�̃�,𝐺 (a, b) (47) 

(iv) 𝑟𝑢,𝐺{𝑣,𝜔} 
′ (a, b) > 𝑟𝑢,𝐺 (a, b)  (48) 

(v) 𝑟�̃�,𝐺{𝑣,𝜔} 
′ (a, b) > 𝑟�̃�,𝐺 (a, b) (49) 

With  𝑟
𝑡𝐴

𝑙
,𝐺{𝑣,𝜔} 

′ (a, b) < 𝑟𝑡𝐴
𝑙

,𝐺
(a, b),  𝑟𝑡𝐴

𝑢
,𝐺{𝑣,𝜔} 

′ (a, b) < 𝑟𝑡𝐴
𝑢

,𝐺 (a, b),   

𝑟
𝑐𝐴

𝑙
,𝐺{𝑣,𝜔} 

′ (a, b) > 𝑟𝑐𝐴
𝑙

,𝐺
(a, b),  𝑟𝑐𝐴

𝑢
,𝐺{𝑣,𝜔} 

′ (a, b) > 𝑟𝑐𝐴
𝑢

,𝐺 (a, b),   

𝑟
𝑔𝐴

𝑙
,𝐺{𝑣,𝜔}

′ (a, b) > 𝑟𝑔𝐴
𝑙

,𝐺
(a, b),  𝑟𝑔𝐴

𝑢
,𝐺{𝑣,𝜔}

′ (a, b) > 𝑟𝑔𝐴
𝑢

,𝐺 (a, b), 

𝑟
𝑢𝐴

𝑙
,𝐺{𝑣,𝜔} 

′ (a, b) > 𝑟𝑢𝐴
𝑙

,𝐺
(a, b),  𝑟𝑢𝐴

𝑢
,𝐺{𝑣,𝜔} 

′ (a, b) > 𝑟𝑢𝐴
𝑢

,𝐺 (a, b) and 

𝑟
𝑓𝐴

𝑙
,𝐺{𝑣,𝜔} 

′ (a, b) > 𝑟𝑓𝐴
𝑙

,𝐺
(a, b),  𝑟𝑓𝐴

𝑢
,𝐺{𝑣,𝜔} 

′ (a, b) > 𝑟𝑓𝐴
𝑢

,𝐺 (a, b). 

In particular, if all of the conditions (i)–(v) are true for some a,b ∈ V , then {𝑣, 𝜔} is 
said to be a strong interval valued pentapartitioned neutrosophic bridge (strong-
IVPPNB) in G.  

Definition 3.13. Let G =(A, B, V, E) be an IVPPNG and and let 𝑣 be a vertex in G. 
Denote 𝐺𝑣 

′  as the partial-IVPPNG of G, in which 𝐺𝑣 
′ =(𝐴′, 𝐵′, 𝑉 ′, 𝐸′)  with  𝑉 ′ = 𝑉 −

{ 𝑣} and 𝐸′= E-{{𝑎, 𝑣} : a∈ 𝑉 − { 𝑣} }. Then, 𝑣is said to be an interval valued 
pentapartitioned neutrosophic cut  vertex (IVPPNCV) in G if at least one of the 
following conditions holds for some a,b ∈ V  

(i) 𝑟𝑡,𝐺𝑣 ′ (a, b) < 𝑟𝑡,𝐺 (a, b)  (50) 

(ii) 𝑟𝑐̃,𝐺𝑣 ′ (a, b) > 𝑟𝑐̃,𝐺 (a, b)  (51) 

(iii) 𝑟�̃�,𝐺𝑣 ′ (a, b) > 𝑟�̃�,𝐺 (a, b)  (52) 



 Interval Valued Pentapartitioned Neutrosophic Graphs with an Application to MCDM 
 

79 
 

(iv) 𝑟𝑢,𝐺𝑣 ′ (a, b) > 𝑟𝑢,𝐺 (a, b)   (53) 

(v) 𝑟�̃�,𝐺𝑣 ′ (a, b) > 𝑟�̃�,𝐺 (a, b)  (54) 

With  𝑟
𝑡𝐴

𝑙
,𝐺𝑣 

′ (a, b) < 𝑟𝑡𝐴
𝑙

,𝐺
(a, b),  𝑟𝑡𝐴

𝑢
,𝐺𝑣 

′ (a, b) < 𝑟𝑡𝐴
𝑢

,𝐺 (a, b),   

𝑟
𝑐𝐴

𝑙
,𝐺𝑣 

′ (a, b) > 𝑟𝑐𝐴
𝑙

,𝐺
(a, b),  𝑟𝑐𝐴

𝑢
,𝐺𝑣 

′  (a, b) > 𝑟𝑐𝐴
𝑢

,𝐺 (a, b),   

𝑟
𝑔𝐴

𝑙
,𝐺𝑣 

′ (a, b) > 𝑟𝑔𝐴
𝑙

,𝐺
(a, b),  𝑟𝑔𝐴

𝑢
,𝐺𝑣 

′ (a, b) > 𝑟𝑔𝐴
𝑢

,𝐺 (a, b), 

𝑟
𝑢𝐴

𝑙
,𝐺𝑣 

′ (a, b) > 𝑟𝑢𝐴
𝑙

,𝐺
(a, b),  𝑟𝑢𝐴

𝑢
,𝐺𝑣 

′ (a, b) > 𝑟𝑢𝐴
𝑢

,𝐺 (a, b) and 

𝑟
𝑓𝐴

𝑙
,𝐺𝑣 

′ (a, b) > 𝑟𝑓𝐴
𝑙

,𝐺
(a, b),  𝑟𝑓𝐴

𝑢
,𝐺𝑣 

′ (a, b) > 𝑟𝑓𝐴
𝑢

,𝐺 (a, b).  

In particular, if all of the conditions (i)–(v) are true for some a, b ∈ V, then 𝑣 is said 
to be a strong interval valued pentapartitioned neutrosophic cut vertex (strong-
IVPPNCV) in G.  

Definition 3.15. Suppose that Ĝ= (P, Q) is an SVPN-graph. Then, the size of Ĝ= (P, 
Q), denoted by S(Ĝ) is defined by S(Ĝ)= (ST(Ĝ), SC(Ĝ), SR(Ĝ), SU(Ĝ), SF(Ĝ)), where 

ST(Ĝ)=∑   u≠k TQ(u, k) denotes the T-size of Ĝ= (P, Q);  (55) 

SC(Ĝ)=∑   u≠k CQ(u, k)  denotes the C-size of Ĝ= (P, Q); (56) 

SR(Ĝ)=∑   u≠k RQ(u, k) denotes the R-size of Ĝ= (P, Q); (57) 

SU(Ĝ)=∑   u≠k UQ(u, k) denotes the U-size of Ĝ= (P, Q); (58) 

SF(Ĝ)= ∑   u≠k FQ(u, k) denotes the F-size of Ĝ= (P, Q).  (59) 

4. Application in Decision Making Problem  

Rapid development of the application of the fuzzy decision making in the real-life 
application is remarkable. There are many applications of the implementation of the 
fuzzy graph model in decision making is evident in the recent research articles.  Multi 
attribute decision-making is one of the decision-making methods in which we analyze 
the attributes according to the criteria for the problems chosen and using an algorithm 
the attributes are evaluated and the best attributes will be selected. More models are 
proposed according to each further developments and extensions of the fuzzy theory, 
we have proposed the decision-making method according to the developed concept of 
interval valued Pentapartioned Neutrosohic graphs. The proposed algorithm using the 
interval valued Pentapartionied Neutrosophic graphs is as follows: 

4.1. Algorithm: 

Step 1: Input the attributes 𝐴 = {𝐴1, 𝐴2, … 𝐴𝑚 } and set of criteria 𝐶 = {𝐶1, 𝐶2, … 𝐶𝑛}. 

Step 2: Construct the interval valued Pentapartitioned relation 𝐾 𝑖 = (𝑘𝑙𝑝
(𝑖)

)
𝑚𝑥𝑚

 

where 𝑖 = 1,2, … 𝑛 &  𝑘𝑙𝑝
𝑖 = ([𝑇𝑙𝑝

(𝑖)𝐿
, 𝑇𝑙𝑝

(𝑖)𝑈
], [𝐶𝑙𝑝

(𝑖)𝐿
, 𝐶𝑙𝑝

(𝑖)𝑈
], [𝑈𝑙𝑝

(𝑖)𝐿
, 𝑈𝑙𝑝

(𝑖)𝑈
], [𝐺𝑙𝑝

(𝑖)𝐿
,

𝐺𝑙𝑝
(𝑖)𝑈

], [𝐹𝑙𝑝
(𝑖)𝐿

, 𝐹𝑙𝑝
(𝑖)𝑈

]),   𝑙, 𝑝 = 1,2, … 𝑛. 

Step 3: Compute the resultant adjacency matrix (K) for the attributes under the 
criteria using the intersection of all the interval valued Pentapartitioned relation (𝐾 𝑖) 
as given, 𝐾 = ⋂ 𝐾 𝑖𝑖  



S. Broumi et al./Oper. Res. Eng. Sci. Theor. Appl. 5(3)2022 68-91 
  

80 
 

Step 4: Calculate the score value of resultant adjacency matrix 𝐾 by using score 
function 𝑆𝑖𝑗 . 

𝑆𝑙𝑝 =

𝑇𝑙𝑝
𝐿 +𝑇𝑙𝑝

𝑈
+(1−𝐶𝑙𝑝

𝐿 )+(1−𝐶𝑙𝑝
𝑈

)+(1−𝑈𝑙𝑝
𝐿 )+(1−𝑈𝑙𝑝

𝑈
)+

(1−𝐺𝑙𝑝
𝐿 )+(1−𝐺𝑙𝑝

𝑈
)+(1−𝐹𝑙𝑝

𝐿 )(1−𝐹𝑙𝑝
𝑈

)

5
  (60) 

Step 5: Calculate the choice values 𝐴ℎ = ∑ 𝑆ℎ𝑗𝑗      h, j=1,2,…m  of each alternative. 

Step 6: The final decision is the 𝐴𝑗  with maximum choice value. 

Step 7: If the 𝑗 has more than one maximum value, then any one may be chosen. 

4.2. An Illustrative Example: 

In the fuzzy set and its extensions, the membership and the other membership 
degree function is sufficient but does not satisfy the needed vague categories. Thus, 
the introduction of the Pentapartitioned Neutrosophic set have paved the way for that 
advancement in the application area. It has five membership degree which mostly 
consider the vagueness of the real-life situations. The introduction of Interval valued 
Pentapartitioned Neutrosophic set is like a cream on top of the cake which adds more 
fuzziness in the conditions and criteria.  

Now we consider the decision-making problems to choose the best international 
airline to take up the journey. The proposed decision-making method based on the 
interval valued Pentapartitioned Neutrosophic set and graph is used to solve this 
decision-making problem. The proposed method is more effective than the other 
proposed models due its membership degrees. Let the attributes be 𝐴 = {𝐴1, 𝐴2, … 𝐴6} 
where 𝐴𝑖 ’s are the Airlines which we consider in our list. The selection of the best 
airline is based the criteria’s 𝐶 = {𝐶1, 𝐶2, … 𝐶5}. where the criteria’s are 

𝐶1 = Safe air travel for passengers during COVID-19 Airline safety rating. 

𝐶2 = High standards of Airport. 

𝐶3 = Onboard product with excellent standards of staff service delivery 

𝐶4 = Specialist Intelligence and Save time & Resources 

𝐶5 = Trusted Experience and Efficient workflows 

Using the proposed method, we input the attributes and the criteria’s. According to 
each criteria Interval valued Pentapartitioned Neutrosophic relation is developed (𝐾𝑖 ) 
as directed graphs. Then the intersection of all the criteria’s are calculated and the 
resultant matrix is given in a graph. Then the score values are calculated for each 
alternative and framed into a graph structure. The choice values are found for each 
attribute and arranged according to find the optimal alternative. The method of 
solving the selected problem is as follows.: 

Step 1: The attributes 𝐴 = {𝐴1, 𝐴2, … 𝐴6} and criteria are 𝐶 = {𝐶1, 𝐶2, … 𝐶6}. 

Step 2: The interval valued Pentapartitioned relation 𝐾 𝑖 (𝑖 = 1, 2, 3, 4, 5) is 
developed for each criteria as in the tables 2 to 6. 

 



 Interval Valued Pentapartitioned Neutrosophic Graphs with an Application to MCDM 
 

81 
 

Table 2. Interval Valued Pentapartitioned Neutrosophic relation (K1) for Criteria 1 
K1 A1 A2 A3 A4 A5 A6 
A1 ([0.3,0.7], 

[0.2,0.6], 
[0.4,0.5], 
[0.1,0.4], 
[0.2,0.6]) 

([0.1,0.5], 
[0.2,0.7], 
[0.3,0.8], 
[0.5,0.9], 
[0.1,0.4]) 

([0.1,0.5], 
[0.2,0.8], 
[0.3,0.6], 
[0.4,0.7], 
[0.5,0.8]) 

([0.2,0.4], 
[0.3,0.5], 
[0.2,0.8], 
[0.1,0.7], 
[0.3,0.6]) 

([0.4,0.8], 
[0.3,0.6], 
[0.3,0.7], 
[0.2,0.6], 
[0.1,0.6]) 

([0.2,0.7], 
[0.1,0.4], 
[0.3,0.7], 
[0.6,0.9], 
[0.4,0.6]) 

A2 ([0.1,0.5], 
[0.2,0.4], 
[0.2,0.6], 
[0.3,0.7], 
[0.4,0.8]) 

([0.3,0.7], 
[0.2,0.6], 
[0.4,0.5], 
[0.1,0.4], 
[0.2,0.6]) 

([0.2,0.6], 
[0.3,0.7], 
[0.4,0.8], 
[0.2,0.7], 
[0.1,0.6]) 

([0.1,0.6], 
[0.2,0.5], 
[0.3,0.7], 
[0.4,0.9], 
[0.5,0.8]) 

([0.2,0.6], 
[0.4,0.8], 
[0.1,0.6], 
[0.3,0.7], 
[0.5,0.9]) 

([0.4,0.8], 
[0.1,0.7], 
[0.5,0.8], 
[0.3,0.6], 
[0.2,0.6]) 

A3 ([0.3,0.6], 
[0.4,0.7], 
[0.1,0.5], 
[0.2,0.8], 
[0.1,0.6]) 

([0.1,0.5], 
[0.2,0.8], 
[0.3,0.6], 
[0.4,0.7], 
[0.1,0.7]) 

([0.3,0.7], 
[0.2,0.6], 
[0.4,0.5], 
[0.1,0.4], 
[0.2,0.6]) 

([0.5,0.9], 
[0.3,0.8], 
[0.2,0.6], 
[0.1,0.5], 
[0.3,0.7]) 

([0.3,0.8], 
[0.2,0.7], 
[0.5,0.9], 
[0.4,0.8], 
[0.1,0.5]) 

([0.1,0.5], 
[0.2,0.6], 
[0.3,0.7], 
[0.4,0.8], 
[0.5,0.8]) 

A4 ([0.2,0.5], 
[0.3,0.7], 
[0.4,0.9], 
[0.5,0.9], 
[0.1,0.7]) 

([0.1,0.6], 
[0.2,0.7], 
[0.3,0.8], 
[0.5,0.9], 
[0.2,0.8]) 

([0.2,0.8], 
[0.3,0.9], 
[0.5,0.7], 
[0.3,0.6], 
[0.1,0.4]) 

([0.3,0.7], 
[0.2,0.6], 
[0.4,0.5], 
[0.1,0.4], 
[0.2,0.6]) 

([0.3,0.9], 
[0.1,0.4], 
[0.4,0.6], 
[0.6,0.9], 
[0.2,0.8]) 

([0.3,0.6], 
[0.4,0.7], 
[0.2,0.5], 
[0.2,0.8], 
[0.1,0.4]) 

A5 ([0.2,0.6], 
[0.1,0.5], 
[0.3,0.7], 
[0.4,0.7], 
[0.1,0.4]) 

([0.1,0.4], 
[0.3,0.7], 
[0.2,0.6], 
[0.4,0.9], 
[0.1,0.7]) 

([0.5,0.9], 
[0.4,0.7], 
[0.3,0.6], 
[0.1,0.7], 
[0.2,0.8]) 

([0.3,0.9], 
[0.5,0.8], 
[0.1,0.6], 
[0.2,0.7], 
[0.4,0.8]) 

([0.3,0.7], 
[0.2,0.6], 
[0.4,0.5], 
[0.1,0.4], 
[0.2,0.6]) 

([0.2,0.6], 
[0.4,0.7], 
[0.2,0.5], 
[0.2,0.8], 
[0.1,0.4]) 

A6 ([0.2,0.6], 
[0.3,0.5], 
[0.3,0.8], 
[0.4,0.9], 
[0.3,0.7]) 

([0.2,0.8], 
[0.3,0.6], 
[0.4,0.8], 
[0.5,0.7], 
[0.2,0.9]) 

([0.6,0.8], 
[0.5,0.7], 
[0.3,0.6], 
[0.2,0.7], 
[0.1,0.4]) 

([0.3,0.6], 
[0.2,0.8], 
[0.5,0.7], 
[0.4,0.9], 
[0.3,0.8]) 

([0.1,0.6], 
[0.3,0.8], 
[0.4,0.7], 
[0.1,0.8], 
[0.2,0.9]) 

([0.3,0.7], 
[0.2,0.6], 
[0.4,0.5], 
[0.1,0.4], 
[0.2,0.6]) 

Table 3. Interval Valued Pentapartitioned Neutrosophic relation (K2) for Criteria 2 
K2 A1 A2 A3 A4 A5 A6 
A1 ([0.2,0.5], 

[0.3,0.7], 
[0.4,0.9], 
[0.5,0.9], 
[0.1,0.7]) 

([0.1,0.5], 
[0.2,0.7], 
[0.3,0.8], 
[0.5,0.9], 
[0.1,0.4]) 

([0.1,0.5], 
[0.2,0.8], 
[0.3,0.6], 
[0.4,0.7], 
[0.1,0.7]) 

([0.1,0.6], 
[0.2,0.7], 
[0.3,0.8], 
[0.5,0.9], 
[0.2,0.8]) 

([0.1,0.4], 
[0.3,0.7], 
[0.2,0.6], 
[0.4,0.9], 
[0.1,0.7]) 

([0.2,0.8], 
[0.3,0.6], 
[0.4,0.8], 
[0.5,0.7], 
[0.2,0.9]) 

A2 ([0.1,0.6], 
[0.2,0.7], 
[0.3,0.8], 
[0.5,0.9], 
[0.2,0.8]) 

([0.2,0.5], 
[0.3,0.7], 
[0.4,0.9], 
[0.5,0.9], 
[0.1,0.7]) 

([0.2,0.6], 
[0.3,0.7], 
[0.4,0.8], 
[0.2,0.7], 
[0.1,0.5]) 

([0.2,0.7], 
[0.3,0.5], 
[0.4,0.8], 
[0.5,0.9], 
[0.4,0.6]) 

([0.2,0.6], 
[0.4,0.8], 
[0.1,0.6], 
[0.3,0.7], 
[0.5,0.9]) 

([0.4,0.8], 
[0.1,0.7], 
[0.5,0.8], 
[0.3,0.6], 
[0.2,0.6]) 

A3 ([0.2,0.8], 
[0.3,0.9], 
[0.5,0.7], 
[0.2,0.6], 

([0.3,0.6], 
[0.4,0.7], 
[0.1,0.5], 
[0.2,0.8], 

([0.2,0.5], 
[0.3,0.7], 
[0.4,0.9], 
[0.5,0.9], 

([0.5,0.9], 
[0.3,0.8], 
[0.3,0.7], 
[0.2,0.8], 

([0.4,0.7], 
[0.5,0.8], 
[0.3,0.6], 
[0.1,0.4], 

([0.4,0.9], 
[0.5,0.8], 
[0.6,0.9], 
[0.3,0.7], 



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82 
 

[0.1,0.4]) [0.1,0.6]) [0.1,0.7]) [0.4,0.9]) [0.2,0.5]) [0.2,0.5]) 
A4 ([0.3,0.7], 

[0.2,0.6], 
[0.4,0.5], 
[0.1,0.4], 
[0.2,0.6]) 

([0.1,0.5], 
[0.2,0.8], 
[0.3,0.6], 
[0.4,0.7], 
[0.1,0.7]) 

([0.5,0.9], 
[0.3,0.8], 
[0.2,0.6], 
[0.1,0.5], 
[0.3,0.7]) 

([0.2,0.5], 
[0.3,0.7], 
[0.4,0.9], 
[0.5,0.9], 
[0.1,0.7]) 

([0.3,0.8], 
[0.2,0.7], 
[0.5,0.9], 
[0.4,0.8], 
[0.2,0.5]) 

([0.2,0.6], 
[0.1,0.5], 
[0.3,0.7], 
[0.5,0.8], 
[0.6,0.9]) 

A5 ([0.3,0.9], 
[0.1,0.4], 
[0.4,0.6], 
[0.6,0.9], 
[0.2,0.8]) 

([0.4,0.7], 
[0.2,0.6], 
[0.1,0.4], 
[0.3,0.8], 
[0.5,0.8]) 

([0.6,0.8], 
[0.6,0.9], 
[0.5,0.7], 
[0.4,0.6], 
[0.1,0.3]) 

([0.7,0.9], 
[0.5,0.8], 
[0.4,0.6], 
[0.3,0.5], 
[0.2,0.7]) 

([0.2,0.5], 
[0.3,0.7], 
[0.4,0.9], 
[0.5,0.9], 
[0.1,0.7]) 

([0.5,0.7], 
[0.4,0.6], 
[0.3,0.6], 
[0.2,0.5], 
[0.1,0.6]) 

A6 ([0.3,0.6], 
[0.4,0.7], 
[0.2,0.5], 
[0.2,0.8], 
[0.1,0.4]) 

([0.4,0.7], 
[0.3,0.5], 
[0.2,0.6], 
[0.3,0.8], 
[0.5,0.9]) 

([0.5,0.8], 
[0.3,0.4], 
[0.4,0.7], 
[0.3,0.6], 
[0.2,0.6]) 

([0.3,0.5], 
[0.2,0.6], 
[0.5,0.8], 
[0.4,0.9], 
[0.6,0.9]) 

([0.4,0.6], 
[0.3,0.5], 
[0.2,0.8], 
[0.1,0.5], 
[0.6,0.9]) 

([0.2,0.5], 
[0.3,0.7], 
[0.4,0.9], 
[0.5,0.9], 
[0.1,0.7]) 

Table 4. Interval Valued Pentapartitioned Neutrosophic relation (K3) for Criteria 3 
K3 A1 A2 A3 A4 A5 A6 
A1 ([0.3,0.6], 

[0.2,0.8], 
[0.5,0.7], 
[0.4,0.9], 
[0.3,0.8]) 

([0.5,0.9], 
[0.3,0.8], 
[0.2,0.6], 
[0.1,0.5], 
[0.3,0.7]) 

([0.1,0.5], 
[0.2,0.8], 
[0.3,0.6], 
[0.4,0.7], 
[0.1,0.7]) 

([0.3,0.7], 
[0.2,0.6], 
[0.4,0.5], 
[0.1,0.4], 
[0.2,0.8]) 

([0.4,0.6], 
[0.3,0.5], 
[0.2,0.8], 
[0.1,0.5], 
[0.6,0.9]) 

([0.5,0.7], 
[0.4,0.6], 
[0.3,0.6], 
[0.2,0.5], 
[0.1,0.6]) 

A2 ([0.5,0.9], 
[0.3,0.8], 
[0.3,0.7], 
[0.2,0.8], 
[0.4,0.9]) 

([0.3,0.6], 
[0.2,0.8], 
[0.5,0.7], 
[0.4,0.9], 
[0.3,0.8]) 

([0.6,0.8], 
[0.6,0.9], 
[0.5,0.7], 
[0.4,0.6], 
[0.1,0.3]) 

([0.1,0.6], 
[0.2,0.7], 
[0.3,0.8], 
[0.5,0.9], 
[0.2,0.5]) 

([0.3,0.9], 
[0.1,0.4], 
[0.4,0.6], 
[0.6,0.9], 
[0.2,0.8]) 

([0.4,0.8], 
[0.1,0.7], 
[0.5,0.8], 
[0.3,0.6], 
[0.2,0.6]) 

A3 ([0.3,0.9], 
[0.1,0.4], 
[0.4,0.6], 
[0.6,0.9], 
[0.2,0.8]) 

([0.4,0.7], 
[0.5,0.8], 
[0.3,0.6], 
[0.1,0.4], 
[0.2,0.5]) 

([0.3,0.6], 
[0.2,0.8], 
[0.5,0.7], 
[0.4,0.9], 
[0.3,0.8]) 

([0.5,0.8], 
[0.3,0.4], 
[0.4,0.7], 
[0.3,0.6], 
[0.2,0.5]) 

([0.1,0.4], 
[0.3,0.7], 
[0.2,0.6], 
[0.4,0.9], 
[0.1,0.7]) 

([0.3,0.6], 
[0.4,0.7], 
[0.2,0.5], 
[0.2,0.8], 
[0.1,0.4]) 

A4 ([0.4,0.7], 
[0.2,0.6], 
[0.1,0.4], 
[0.3,0.8], 
[0.5,0.8]) 

([0.3,0.6], 
[0.4,0.7], 
[0.1,0.5], 
[0.2,0.8], 
[0.1,0.6]) 

([0.4,0.9], 
[0.5,0.8], 
[0.6,0.9], 
[0.5,0.7], 
[0.2,0.5]) 

([0.3,0.6], 
[0.2,0.8], 
[0.5,0.7], 
[0.4,0.9], 
[0.3,0.8]) 

([0.5,0.9], 
[0.3,0.8], 
[0.3,0.7], 
[0.2,0.8], 
[0.4,0.9]) 

([0.2,0.8], 
[0.3,0.6], 
[0.4,0.8], 
[0.5,0.7], 
[0.2,0.9]) 

A5 ([0.6,0.8], 
[0.6,0.9], 
[0.5,0.7], 
[0.4,0.6], 
[0.1,0.3]) 

([0.3,0.7], 
[0.2,0.6], 
[0.4,0.5], 
[0.1,0.4], 
[0.2,0.8]) 

([0.1,0.5], 
[0.2,0.8], 
[0.3,0.6], 
[0.4,0.7], 
[0.1,0.7]) 

([0.2,0.5], 
[0.3,0.7], 
[0.4,0.9], 
[0.5,0.8], 
[0.1,0.7]) 

([0.3,0.6], 
[0.2,0.8], 
[0.5,0.7], 
[0.4,0.9], 
[0.3,0.8]) 

([0.7,0.9], 
[0.5,0.8], 
[0.4,0.6], 
[0.3,0.5], 
[0.2,0.7]) 

A6 ([0.7,0.9], 
[0.5,0.8], 
[0.4,0.6], 
[0.3,0.5], 
[0.2,0.7]) 

([0.3,0.9], 
[0.1,0.4], 
[0.4,0.6], 
[0.6,0.9], 
[0.2,0.8]) 

([0.3,0.6], 
[0.4,0.7], 
[0.2,0.5], 
[0.2,0.8], 
[0.1,0.4]) 

([0.4,0.7], 
[0.2,0.6], 
[0.1,0.4], 
[0.3,0.8], 
[0.5,0.8]) 

([0.1,0.5], 
[0.2,0.7], 
[0.3,0.8], 
[0.5,0.9], 
[0.1,0.4]) 

([0.3,0.6], 
[0.2,0.8], 
[0.5,0.7], 
[0.4,0.9], 
[0.3,0.8]) 



 Interval Valued Pentapartitioned Neutrosophic Graphs with an Application to MCDM 
 

83 
 

Table 5. Interval Valued Pentapartitioned Neutrosophic relation (K4) for Criteria 4 
K4 A1 A2 A3 A4 A5 A6 
A1 ([0.3,0.5], 

[0.2,0.7], 
[0.5,0.9], 
[0.4,0.8], 
[0.3,0.8]) 

([0.5,0.9], 
[0.3,0.8], 
[0.3,0.7], 
[0.2,0.8], 
[0.4,0.9]) 

([0.3,0.9], 
[0.1,0.4], 
[0.4,0.6], 
[0.6,0.9], 
[0.2,0.8]) 

([0.4,0.7], 
[0.2,0.6], 
[0.1,0.4], 
[0.3,0.8], 
[0.5,0.8]) 

([0.6,0.8], 
[0.6,0.9], 
[0.5,0.7], 
[0.4,0.6], 
[0.1,0.3]) 

([0.7,0.9], 
[0.5,0.8], 
[0.4,0.6], 
[0.3,0.5], 
[0.2,0.7]) 

A2 ([0.5,0.9], 
[0.3,0.8], 
[0.2,0.6], 
[0.1,0.5], 
[0.3,0.7]) 

([0.3,0.5], 
[0.2,0.7], 
[0.5,0.9], 
[0.4,0.8], 
[0.3,0.8]) 

([0.4,0.7], 
[0.5,0.8], 
[0.3,0.6], 
[0.1,0.4], 
[0.2,0.5]) 

([0.3,0.6], 
[0.4,0.7], 
[0.1,0.5], 
[0.2,0.8], 
[0.1,0.6]) 

([0.3,0.7], 
[0.2,0.6], 
[0.4,0.5], 
[0.1,0.4], 
[0.2,0.8]) 

([0.3,0.9], 
[0.1,0.4], 
[0.4,0.6], 
[0.6,0.9], 
[0.2,0.8]) 

A3 ([0.1,0.5], 
[0.2,0.8], 
[0.3,0.6], 
[0.4,0.7], 
[0.1,0.7]) 

([0.6,0.8], 
[0.6,0.9], 
[0.5,0.7], 
[0.4,0.6], 
[0.1,0.3]) 

([0.3,0.5], 
[0.2,0.7], 
[0.5,0.9], 
[0.4,0.8], 
[0.3,0.8]) 

([0.4,0.9], 
[0.5,0.8], 
[0.6,0.9], 
[0.5,0.7], 
[0.2,0.5]) 

([0.1,0.3], 
[0.2,0.8], 
[0.3,0.7], 
[0.4,0.7], 
[0.1,0.5]) 

([0.3,0.6], 
[0.4,0.7], 
[0.2,0.5], 
[0.2,0.8], 
[0.1,0.4]) 

A4 ([0.3,0.7], 
[0.2,0.6], 
[0.4,0.5], 
[0.1,0.4], 
[0.2,0.8]) 

([0.1,0.6], 
[0.2,0.7], 
[0.3,0.8], 
[0.5,0.9], 
[0.2,0.5]) 

([0.5,0.8], 
[0.3,0.4], 
[0.4,0.7], 
[0.3,0.6], 
[0.2,0.5]) 

([0.3,0.5], 
[0.2,0.7], 
[0.5,0.9], 
[0.4,0.8], 
[0.3,0.8]) 

([0.2,0.5], 
[0.3,0.7], 
[0.4,0.9], 
[0.5,0.8], 
[0.1,0.6]) 

([0.4,0.7], 
[0.2,0.6], 
[0.1,0.4], 
[0.3,0.8], 
[0.5,0.8]) 

A5 ([0.4,0.6], 
[0.3,0.5], 
[0.2,0.8], 
[0.1,0.4], 
[0.6,0.9]) 

([0.3,0.9], 
[0.1,0.4], 
[0.4,0.6], 
[0.6,0.9], 
[0.2,0.8]) 

([0.1,0.4], 
[0.3,0.7], 
[0.2,0.6], 
[0.4,0.9], 
[0.1,0.7]) 

([0.5,0.9], 
[0.3,0.8], 
[0.3,0.7], 
[0.2,0.8], 
[0.4,0.6]) 

([0.3,0.5], 
[0.2,0.7], 
[0.5,0.9], 
[0.4,0.8], 
[0.3,0.8]) 

([0.2,0.7], 
[0.1,0.5], 
[0.3,0.8], 
[0.5,0.9], 
[0.1,0.4]) 

A6 ([0.5,0.7], 
[0.4,0.6], 
[0.3,0.6], 
[0.2,0.5], 
[0.1,0.6]) 

([0.4,0.8], 
[0.1,0.7], 
[0.5,0.8], 
[0.3,0.6], 
[0.2,0.5]) 

([0.3,0.6], 
[0.4,0.7], 
[0.2,0.5], 
[0.2,0.8], 
[0.1,0.4]) 

([0.2,0.8], 
[0.3,0.6], 
[0.4,0.8], 
[0.5,0.7], 
[0.2,0.9]) 

([0.7,0.9], 
[0.5,0.8], 
[0.4,0.6], 
[0.3,0.5], 
[0.2,0.7]) 

([0.3,0.5], 
[0.2,0.7], 
[0.5,0.9], 
[0.4,0.8], 
[0.3,0.8]) 

Table 6. Interval Valued Pentapartitioned Neutrosophic relation (K5) for Criteria 5 
K5 A1 A2 A3 A4 A5 A6 
A1 ([0.5,0.9], 

[0.3,0.8], 
[0.4,0.7], 
[0.2,0.6], 
[0.3,0.5]) 

([0.3,0.7], 
[0.2,0.5], 
[0.1,0.4], 
[0.4,0.8], 
[0.3,0.6]) 

([0.1,0.5], 
[0.2,0.8], 
[0.3,0.6], 
[0.4,0.7], 
[0.2,0.6]) 

([0.3,0.7], 
[0.2,0.6], 
[0.4,0.5], 
[0.1,0.4], 
[0.2,0.8]) 

([0.4,0.6], 
[0.3,0.5], 
[0.2,0.8], 
[0.1,0.4], 
[0.6,0.9]) 

([0.5,0.7], 
[0.4,0.6], 
[0.3,0.6], 
[0.2,0.5], 
[0.1,0.6]) 

A2 ([0.4,0.8], 
[0.3,0.5], 
[0.4,0.6], 
[0.2,0.5], 
[0.3,0.9]) 

([0.5,0.9], 
[0.3,0.8], 
[0.4,0.7], 
[0.2,0.6], 
[0.3,0.5]) 

([0.6,0.8], 
[0.6,0.9], 
[0.5,0.7], 
[0.4,0.6], 
[0.2,0.4]) 

([0.1,0.6], 
[0.2,0.7], 
[0.3,0.8], 
[0.5,0.9], 
[0.2,0.5]) 

([0.3,0.9], 
[0.1,0.4], 
[0.4,0.6], 
[0.6,0.9], 
[0.1,0.5]) 

([0.4,0.8], 
[0.1,0.7], 
[0.5,0.8], 
[0.3,0.6], 
[0.2,0.5]) 

A3 ([0.3,0.9], 
[0.1,0.4], 
[0.4,0.6], 
[0.6,0.9], 

([0.4,0.7], 
[0.5,0.8], 
[0.3,0.6], 
[0.1,0.4], 

([0.5,0.9], 
[0.3,0.8], 
[0.4,0.7], 
[0.2,0.6], 

([0.5,0.8], 
[0.3,0.4], 
[0.4,0.7], 
[0.3,0.6], 

([0.1,0.4], 
[0.3,0.7], 
[0.2,0.6], 
[0.4,0.9], 

([0.3,0.6], 
[0.4,0.7], 
[0.2,0.5], 
[0.2,0.8], 



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[0.2,0.8]) [0.2,0.5]) [0.3,0.5]) [0.2,0.5]) [0.1,0.5]) [0.1,0.4]) 
A4 ([0.4,0.7], 

[0.2,0.6], 
[0.1,0.4], 
[0.3,0.8], 
[0.5,0.8]) 

([0.3,0.6], 
[0.4,0.7], 
[0.1,0.5], 
[0.2,0.8], 
[0.1,0.6]) 

([0.4,0.9], 
[0.5,0.8], 
[0.6,0.9], 
[0.5,0.7], 
[0.2,0.5]) 

([0.5,0.9], 
[0.3,0.8], 
[0.4,0.7], 
[0.2,0.6], 
[0.3,0.5]) 

([0.5,0.9], 
[0.3,0.8], 
[0.3,0.7], 
[0.2,0.8], 
[0.4,0.6]) 

([0.2,0.8], 
[0.3,0.6], 
[0.4,0.8], 
[0.5,0.7], 
[0.2,0.9]) 

A5 ([0.6,0.8], 
[0.6,0.9], 
[0.5,0.7], 
[0.4,0.6], 
[0.1,0.3]) 

([0.3,0.7], 
[0.2,0.6], 
[0.4,0.5], 
[0.1,0.4], 
[0.2,0.8]) 

([0.1,0.3], 
[0.2,0.8], 
[0.3,0.7], 
[0.4,0.7], 
[0.6,0.9]) 

([0.2,0.5], 
[0.3,0.7], 
[0.4,0.9], 
[0.5,0.8], 
[0.1,0.6]) 

([0.5,0.9], 
[0.3,0.8], 
[0.4,0.7], 
[0.2,0.6], 
[0.3,0.5]) 

([0.7,0.9], 
[0.5,0.8], 
[0.4,0.6], 
[0.3,0.5], 
[0.2,0.7]) 

A6 ([0.7,0.9], 
[0.5,0.8], 
[0.4,0.6], 
[0.6,0.9], 
[0.2,0.8]) 

([0.3,0.9], 
[0.1,0.4], 
[0.4,0.6], 
[0.6,0.9], 
[0.2,0.8]) 

([0.3,0.6], 
[0.4,0.7], 
[0.2,0.5], 
[0.2,0.8], 
[0.1,0.4]) 

([0.4,0.7], 
[0.2,0.6], 
[0.1,0.4], 
[0.3,0.8], 
[0.5,0.9]) 

([0.2,0.7], 
[0.1,0.5], 
[0.3,0.8], 
[0.5,0.9], 
[0.1,0.4]) 

([0.5,0.9], 
[0.3,0.8], 
[0.4,0.7], 
[0.2,0.6], 
[0.3,0.5]) 

 

Figure 5: Interval Valued Pentapartitioned Neutrosophic directed graph structure for 
𝐶𝑖 (𝑖 = 1, 2, 3, 4, 5) 

Step 3: From the interval valued Pentapartitioned relation 𝐾 𝑖 (𝑖 = 1, 2, 3, 4, 5), 
computing the intersection and find the resultant matrix as in Table 7. 

 



 Interval Valued Pentapartitioned Neutrosophic Graphs with an Application to MCDM 
 

85 
 

Table 7. The resultant matrix 
K A1 A2 A3 A4 A5 A6 
A1 ([0.2,0.5], 

[0.3,0.8], 
[0.5,0.9], 
[0.5,0.9], 
[0.3,0.8]) 

([0.1,0.5], 
[0.3,0.8], 
[0.3,0.8], 
[0.5,0.9], 
[0.4,0.9]) 

([0.1,0.5], 
[0.2,0.8], 
[0.4,0.6], 
[0.6,0.9], 
[0.5,0.8]) 

([0.1,0.4], 
[0.3,0.7], 
[0.4,0.8], 
[0.5,0.9], 
[0.5,0.8]) 

([0.1,0.4], 
[0.6,0.9], 
[0.5,0.8], 
[0.4,0.9], 
[0.6,0.9]) 

([0.2,0.7], 
[0.5,0.8], 
[0.4,0.8], 
[0.6,0.9], 
[0.4,0.9]) 

A2 ([0.1,0.5], 
[0.3,0.8], 
[0.4,0.8], 
[0.5,0.9], 
[0.4,0.9]) 

([0.2,0.5], 
[0.3,0.8], 
[0.5,0.9], 
[0.5,0.9], 
[0.3,0.8]) 

([0.2,0.6], 
[0.6,0.9], 
[0.5,0.8], 
[0.4,0.7], 
[0.2,0.6]) 

([0.1,0.6], 
[0.3,0.8], 
[0.4,0.8], 
[0.5,0.9], 
[0.5,0.8]) 

([0.2,0.6], 
[0.4,0.8], 
[0.4,0.6], 
[0.6,0.9], 
[0.5,0.9]) 

([0.3,0.8], 
[0.1,0.7], 
[0.5,0.8], 
[0.6,0.9], 
[0.2,0.8]) 

A3 ([0.1,0.5], 
[0.4,0.9], 
[0.5,0.7], 
[0.6,0.9], 
[0.2,0.8]) 

([0.1,0.5], 
[0.6,0.9], 
[0.5,0.7], 
[0.4,0.8], 
[0.2,0.7]) 

([0.2,0.5], 
[0.3,0.8], 
[0.5,0.9], 
[0.5,0.9], 
[0.3,0.8]) 

([0.4,0.8], 
[0.5,0.8], 
[0.6,0.9], 
[0.5,0.8], 
[0.4,0.9]) 

([0.1,0.3], 
[0.5,0.8], 
[0.5,0.9], 
[0.4,0.9], 
[0.2,0.7]) 

([0.1,0.5], 
[0.5,0.8], 
[0.6,0.9], 
[0.5,0.7], 
[0.5,0.8]) 

A4 ([0.2,0.5], 
[0.3,0.7], 
[0.4,0.9], 
[0.5,0.9], 
[0.5,0.8]) 

([0.1,0.5], 
[0.4,0.8], 
[0.3,0.8], 
[0.5,0.9], 
[0.2,0.8]) 

([0.2,0.8], 
[0.5,0.9], 
[0.6,0.9], 
[0.5,0.7], 
[0.3,0.7]) 

([0.2,0.5], 
[0.3,0.8], 
[0.5,0.9], 
[0.5,0.9], 
[0.3,0.8]) 

([0.2,0.5], 
[0.3,0.8], 
[0.5,0.9], 
[0.6,0.9], 
[0.4,0.9]) 

([0.2,0.6], 
[0.4,0.7], 
[0.4,0.8], 
[0.5,0.8], 
[0.6,0.9]) 

A5 ([0.2,0.6], 
[0.6,0.9], 
[0.5,0.8], 
[0.6,0.9], 
[0.6,0.9]) 

([0.1,0.4], 
[0.3,0.7], 
[0.4,0.6], 
[0.6,0.9], 
[0.5,0.8]) 

([0.1,0.3], 
[0.6,0.9], 
[0.5,0.7], 
[0.4,0.9], 
[0.6,0.9]) 

([0.2,0.5], 
[0.5,0.8], 
[0.4,0.9], 
[0.5,0.8], 
[0.4,0.8]) 

([0.2,0.5], 
[0.3,0.8], 
[0.5,0.9], 
[0.5,0.9], 
[0.3,0.8]) 

([0.2,0.6], 
[0.5,0.8], 
[0.4,0.8], 
[0.5,0.9], 
[0.2,0.7]) 

A6 ([0.2,0.6], 
[0.5,0.8], 
[0.4,0.8], 
[0.6,0.9], 
[0.3,0.8]) 

([0.2,0.7], 
[0.3,0.7], 
[0.5,0.8], 
[0.6,0.9], 
[0.5,0.9]) 

([0.3,0.6], 
[0.5,0.7], 
[0.4,0.7], 
[0.3,0.8], 
[0.2,0.6]) 

([0.2,0.5], 
[0.3,0.8], 
[0.5,0.8], 
[0.5,0.9], 
[0.6,0.9]) 

([0.1,0.5], 
[0.5,0.8], 
[0.4,0.8], 
[0.5,0.9], 
[0.6,0.9]) 

([0.2,0.5], 
[0.3,0.8], 
[0.5,0.9], 
[0.5,0.9], 
[0.3,0.8]) 

Step 4: By using the score function 𝑆𝑖𝑗 , in the resultant matrix of Table 7, find the 

score matrix as in Table 8. 

Table 8. The score matrix of K and the choice values of the Attributes 𝐴𝑗  

K A1 A2 A3 A4 A5 A6 Choice Value 
A1 0.74 0.74 0.76 0.72 0.58 0.72 4.24 
A2 0.72 0.74 0.82 0.74 0.74 0.9 4.66 
A3 0.72 0.76 0.74 0.68 0.7 0.66 4.26 
A4 0.74 0.78 0.78 0.86 0.68 0.74 4.58 
A5 0.6 0.74 0.58 0.72 0.74 0.76 4.14 
A6 0.74 0.78 0.94 0.68 0.64 0.74 4.52 

 



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86 
 

 

Figure 6. Interval Valued Pentapartitioned Neutrosophic directed graph with score 
values 

Step 5: From the score matrix of Table 8, calculate the choice values for each 
attribute 𝐴𝑖 (𝑖 = 1 𝑡𝑜 6). 

Step 6: Arrange the attributes according to their choice values in descending order, 
the final decision is 𝐴2.  

4.66 > 4.58 > 5.52 > 4.26 > 4.24 > 4.14 

𝐴2 >  𝐴4 >  𝐴6 >  𝐴3 >  𝐴1 >  𝐴5 

By using the proposed decision-making method, the optimal decision is decided. 
By using the method, the alternatives are arranged in descending order and the second 
airline is chosen as the best airline with the condition which satisfies all the criteria. 

5. Discussion 

Interval valued pentapartitioned neutrosophic sets provide a powerful tool to 
represent the information with imprecise and indeterminate data and have fruitful 
applications. Pentapartitioned neutrosophic sets is an extension of the Neutrosophic 
sets with the membership function classified into five categories. The interval valued 
pentapartitioned neutrosophic sets which is introduced is an advancement of the 
pentapartitioned set like each membership representation comes in an interval. In this 



 Interval Valued Pentapartitioned Neutrosophic Graphs with an Application to MCDM 
 

87 
 

paper, the interval valued pentapartitioned neutrosophic sets have been implemented 
on the graph theoretical concepts and interval valued pentapartitioned neutrosophic 
graph have been developed. The significant operations of interval-valued 
pentapartitioned neutrosophic sets and its graphs are investigated. This helps the 
decision makers more sufficient for taking their input best suit to their domain of 
reference. The properties of interval valued pentapartitioned neutrosophic graph as 
cut vertex, bridge, degree are studied and examined with suitable examples. Hence, 
the proposed graphs and their  basic properties have enough capabilities to address 
the related dependability on the indterminate information. A decision-making method 
have been developed using the interval valued pentapartitioned neutrosophic graph 
with a numerical illustration.  

6. Conclusion  

Interval valued pentapartitioned nutrosophic graphs are generalization of 
pentapartitioned nutrosophic graphs and provide a sufficient space for complex 
decision-making situations. In this research paper some properties of interval valued 
pentapartitioned neutrosophic graph as cut vertex, bridge, degree are studied and 
examined with suitable examples. Using the proposed Interval valued 
Pentapartitioned Nutrosophic graphs, a decision-making method has been developed 
and applied in a real-life situation with numerical illustrations. In future, concepts can 
be developed in the interval valued pentapartitioned neutrosophic soft graphs, 
interval valued quadripartitioned neutrosophic graphs, Strong interval valued 
pentapartitioned neutrosophic graphs, etc. Also, one can extend the developed 
concepts into isomorphic properties and regularity properties in the proposed graph 
structures. The Interval valued pentapartitioned neutrosophic graph can be extended 
to regular and irregular interval valued pentapartitioned neutrosophic graph, 
interval-valued pentapartitioned neutrosophic intersection graphs, interval-valued 
pentapartitioned neutrosophic hypergraphs, and so on. The interval-valued 
pentapartitioned neutrosophic graph can be used in modeling the network, 
telecommunication, image processing, computer networks, expert systems…etc. 

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https://doi.org/10.1109/IFSANAFIPS.2013.6608375
https://doi.org/10.1108/MD-01-2022-0120
https://doi.org/10.2307/2272014

	Interval Valued Pentapartitioned Neutrosophic Graphs with an Application to MCDM
	Said Broumi 1,2*, D. Ajay 3, P. Chellamani 3, Lathamaheswari Malayalan 4, Mohamed Talea 1, Assia Bakali 5, Philippe Schweizer 6, Saeid Jafari 7
	1. Introduction
	2. Preliminaries
	3. Interval Pentapartitioned neutrosophic graphs
	4. Application in Decision Making Problem
	4.1. Algorithm:
	4.2. An Illustrative Example:

	5. Discussion
	6. Conclusion
	References: