# A Novel Description of Some Concepts in Interval-Valued Intuitionistic Fuzzy Graph with an Application

Authors: Xiaoli Qiang; Saeed Kosari; Xiang Chen; Ali Asghar Talebi; Ghulam Muhiuddin; Seyed Hossein Sadati
Journal: Advances in Mathematical Physics
 (2022)

---

## Abstract

Covering, matching, and domination are the basic concepts in graphs that play a decisive role in the properties of graphs. Calculating these parameters is one of the difficulties in fuzzy graphs when it is not possible to accurately determine the values of the vertices of a graph. The interval-valued intuitionistic fuzzy graph (IVIFG) is one of the fuzzy graphs which can play an important role in solving uncertain problems in different sciences such as psychology, biological sciences, medicine, and social networks. The necessity of using a range of value instead of one number caused them to help researchers in optimizing and saving time and cost. In this study, we introduce some of the specific concepts such as covering, matching, and paired domination using strong arc or effective edges by giving appropriate examples. In addition, we have calculated strong node covering number, strong independent number, and other parameters of complete bipartite IVIFGs with several examples. Finally, we have presented an application of IVIFG in social networks.


---

## Body

## 1. Introduction

Graphs are an inevitable tool in applied mathematics. Among the various concepts in graphs, some concepts are more important such as covering, matching, and domination. These concepts are closely related to vertices, as one of the most important components of the graph, and cause them to participate in many analyses related to vertices. Many studies have been done by researchers on various graphs. It is difficult to examine these concepts when the exact values for the vertices cannot be considered.In 1965, Zadeh [1] presented the basic idea of fuzzy set (FS) where its prominent feature was the allocation of membership degree between 0 and 1 to each element in a set. Zadeh [2] also introduced the interval-valued fuzzy set (IVFS) in 1975, in which membership degrees were intervals of numbers. Roselfeld [3] defined a new concept called the fuzzy graph (FG) by employing fuzzy relations on FS. FGs were considered by researchers in the fields related to ambiguous and uncertain problems. They were able to find numerous applications in solving and modeling problems in computer science, engineering, system analysis, economics, network routing, transportation, and so on. With the advent of new indefinite problems, it became clear that a membership function could not well express the ambiguity in subjective perceptions and the complexity of data. To overcome this shortcoming of the FS, Atanassov [4] proposed an extension of FS by introducing nonmembership function and defined intuitionistic fuzzy set (IFS). IFGs were first introduced by Atanassov [5] in 1999 and was further discussed in [6]. Mahapatra et al. [7–9] explored concepts on fuzzy graphs. Rashmanlou and Pal [10, 11] studied different kinds of FGs. Kosari et al. [12] presented vague graph structure with an application in medical diagnosis. Kou et al. [13] studied some properties of vague graph (VG). Krishna et al. [14] studied new results in cubic graphs. Talebi et al. [15, 16] defined Cayley-FGs and some operations on level graphs of bipolar FGs. Atanassov [17] recently introduced some new topological operators over IFSs. Mathew et al. [18] conducted research on vertex rough graphs. Some concepts in IVFGs and neutrosophic graphs are studied by Jan et al. [19]. Voskoglou [20] used a combination of soft sets and gray numbers in decision-making. Mahapatra et al. [21–23] introduced concepts of neutrosophic graphs used in social networks.The decision to determine accurate numerical values in uncertain and inaccurate evaluations of information, which often occur in practical situations, is associated with difficulties. Thus, in 1989, Atanassov and Gargov [24] introduced the idea of the interval-valued intuitionistic fuzzy set (IVIFS) in order to unify perceptions and quantify the uncertain nature of the mind. This concept is defined by a membership function, a nonmembership function, and a hesitant function whose values are intervals between 0 and 1 instead of exact numbers. IVIFS has been widely used in many areas, such as decision-making [25], pattern recognition [26], medical diagnosis [27], and graph theory [28]. The concept of interval-valued fuzzy graphs (IVFGs) is presented by Hongmei and Lianhua in [29]. Akram et al. [30, 31] defined certain types of IVFGs. The product of IVIFGs was proposed by Mishra and Pal in [32]. The strong IVIFG concept is described by Ismayil and Ali [33]. Rashmanlou et al. [25, 34–36] studied some IVIFG concepts.The purpose of this paper is to find a way to determine the concepts of vertex covering, matching, and paired domination in IVIFGs where we are dealing with interval-valued numbers instead of fuzzy numbers. The previous definition limitations in the vertex covering, matching, and paired domination of FGs have directed us to offer new classifications in terms of IVIFG. These concepts have already been studied by some researchers in a variety of FGs. Sahoo et al. [37] investigated covering and paired domination in IFGs.The rest of this article is organized as follows: Section2 briefly reviews related basic concepts to IVIFGs. In Section 3, we introduced the concepts of strong vertex covering, independent vertex covering, and perfect strong matching in an IVIFG by strong edges and defined some of its properties in specific types of IVIFGs. In this section, we introduce paired domination in IVIFG and examine its implications. Finally, we present an application of IVIFG on social networks in Section 4.


## 2. Preliminaries

In this section, we briefly define some of the basic concepts for entering the main discussion.Definition 1 (see [24]).
An IVIFSA in X can be described as
(1)A=x,μALx,μAUx,νALx,νAUxx∈X,where 0≤μALx≤μAU≤1, 0≤νALx≤νAU≤1, and 0≤μAUx+νAUx≤1, for all x∈X.
Similarly, the intervalsμALx,μAUx and νALx,νAUx denoted the MV and non-MV of an element x, respectively. If each of the intervals contains only one value for each x∈X, we have
(2)μAx=μALx=μAUx,νAx=νALx=νAUx.
Furthermore, the hesitancy degree of each elementx is as follows:
(3)1−μALx−νALx,1−μAUx−νAUx.Definition 2 (see [33]).
An IVIFG of an underlying graphG∗=V,E is a pair G=V,A,B so that
(4)A=μALx,μAUx,νALx,νAUxx∈V,is an IVIFS in V and
(5)B=μBLxy,μBUxy,νBLxy,νBUxyxy∈E,is an interval-valued intuitionistic fuzzy relation (IVIFR) V×V so that
(6)μB:E⊆V×V⟶D0,1,νB:E⊆V×V⟶D0,1,μBLxy≤minμALx,μALy,μBUxy≤minμAUx,μAUy,νBLxy≥maxνALx,νALy,νBUxy≥maxνAUx,νAUy,and μBUxy+νBUxy≤1, for each xy∈E.Definition 3 (see [34]).
An edgexy of an IVIFG, G is named a strong arc (SA) or effective edge if
(7)μBLxy=minμALx,μALy,μBUxy=minμAUx,μAUy,νBLxy=maxνALx,νALy,νBUxy=maxνAUx,νAUy.Definition 4 (see [34]).
An IVIFG is complete, if(8)μBLxy=minμALx,μALy,νBLxy=maxνALx,νALy,μBUxy=minμAUx,μAUy,νBUxy=maxνAUx,νAUy,for all xy∈V×V.As a result of the above definition, the following definition can be provided.Definition 5.
An IVIFGG is named bipartite whenever the vertex set V can be partitioned into two nonempty sets V1 and V2 so that μBLxy=μBUxy=0 and νBLxy=νBUxy=0, for xy∈V1 or xy∈V2. If
(9)μBLxy=minμALx,μALy,μBUxy=minμAUx,μAUy,νBLxy=maxνALx,νALy,νBUxy=maxνAUx,νAUy,for all x∈V1 and y∈V2; then, G is named a complete bipartite IVIFG (CB-IVIFG) and is shown by Kσ1,σ2.All the basic notations are shown in Table1.Table 1
Some basic notations.
NotationMeaningIVFSInterval-valued fuzzy setIFGIntuitionistic fuzzy graphIVIFSInterval-valued intuitionistic fuzzy setIVIFGInterval-valued intuitionistic fuzzy graphCIVIFGComplete interval-valued intuitionistic fuzzy graphSAStrong arcCBComplete bipartiteSCStrong coverSCNStrong covering numberSNCStrong node coverISAIncident strong arcSNCNStrong node covering numberIVMBInterval-valued membership boundIVNMBInterval-valued nonmembership boundSIStrong independentSISStrong independent setSINStrong independent numberINIsolated nodeSACStrong arc coverPDPaired dominationSPDNStrong paired domination numberSMStrong matchingSMNStrong matching numberSIACStrong independent arc coverPSMPerfect strong matchingSDSStrong dominating setSPDSStrong paired dominating set


## 3. Covering, Matching, and Paired Domination in the IVIFGs

In this section, we introduce covering, matching, and paired domination in the IVIFGs by the weight of strong edges and examine some of its properties and results.Definition 6.
LetG=V,A,B be an IVIFG. An SNC in an IVIFG G is the set D of nodes that cover all SAs of G. The weight of an SNC D is denoted as
(10)Wnc=WncLμD,WncUμD,WncLνD,WncUνD,Wnc=∑x∈DμBLxy,∑x∈DμBUxy,∑x∈DνBLxy,∑x∈DνBUxy,so that μBLxy and μBUxy are the minimum of the lower and upper of IVMBs and νBLxy and νBUxy are the maximum of the lower and upper of IVNMBs of all SAs incident on x, respectively.
An SNCN of an IVIFGG is shown as follows αs0G=αs0=αs0Lμ,αs0Uμ,αs0Lν,αs0Uν so that
(11)αs0Lμ=minWncLμDistheweightofSNCsofG,αs0Uμ=minWncUμDistheweightofSNCsofG,αs0Lν=maxWncLνDistheweightofSNCsofG,αs0Uν=maxWncUνDistheweightofSNCsofG.
A minimum SNC in an IVIFGG is an SNC of minimum IVMBs and maximum IVNMBs.Theorem 7.
LetG=V,A,B be a CIVIFG. Then,
(12)αs0Lμ=r−1μBLxy,αs0Lν=r−1νBLxy,αs0Uμ=r−1μBUxy,αs0Uν=r−1νBUxy,where μBLxy and μBUxy are the lower and upper of IVMBs and νBLxy and νBUxy are the lower and upper of IVNMBs of the weakest arc in G. Note that r is the number of vertex in G.Proof.
SinceG is a CIVIFG, all arcs are strong, and every node is neighbor to all other vertices. So, any set includes r−1 nodes forming an SNC of G.
Letx be a vertex having minimum of IVMBs and maximum of IVNMBs in G. Suppose y1,y2⋯,yn−1 is the node neighbor to x. Then, the r−1 arcs xy1,xy2,⋯,xyr−1 are all weakest arcs of G, and strength of each arcs is equal to μBLxy,μBUxy,νBLxy,νBUxy, which y∈y1,y2,⋯,yr−1.
Hence, the setD=y1,y2,⋯,yr−1 of r−1 vertices forms an SNC of G with
(13)WncLμD=∑yi∈DμBLxyi=μBLxy1+μBLxy2+⋯+μBLxyr−1,WncUμD=∑yi∈DμBUxyi=μBUxy1+μBUxy2+⋯+μBUxyr−1,where μBLxyi,i=1,2,⋯,r−1 is the minimum lower of IVMB and μBUxyi,i=1,2,⋯,r−1 is the minimum upper of IVNMB of SAs incident on yi. Then,
(14)αs0Lμ=μBLxy+μBLxy+⋯+μBLxy,αs0Uμ=μBUxy+μBUxy+⋯+μBUxy,where μBLxy and μBUxy are the lower and upper of IVMBs of a weakest arc in G.
Hence,αs0Lμ=r−1μBLxy and αs0Uμ=r−1μBUxy. Similarly,
(15)WncLνD=∑yi∈DνBLxyi=νBLxy1+νBLxy2+⋯+νBLxyr−1,WncUνD=∑yi∈DνBUxyi=νBUxy1+νBUxy2+⋯+νBUxyr−1,where νBLxyi and νBUxyi, i=1,2,⋯,r−1 are the maximum lower and upper of IVNMBs of all SAs incident on yi. Then,
(16)αs0Lν=νBLxy+νBLxy+⋯+νBLxy,αs0Uν=νBUxy+νBUxy+⋯+νBUxy,where νBLxy and νBUxy are the lower and upper of IVNMBs of a weakest arc in G. Hence, αs0Lν=r−1νBLxy and αs0Uν=r−1νBUxy.Theorem 8.
For a CB-IVIFGKσ1,σ2 with partite set V1 and V2,
(17)αs0LμKσ1,σ2=minWncLμV1,WncLμV2,αs0UμKσ1,σ2=minWncUμV1,WncUμV2,αs0LνKσ1,σ2=maxWncLνV1,WncLνV2,αs0UνKσ1,σ2=maxWncUνV1,WncUνV2.Proof.
All arcs inKσ1,σ2 are strong, and each node in V1 is neighbor with all nodes in V2 and contrariwise. The set of all arcs of Kσ1,σ2 is a set of all arcs incident on each node of V1 or a set of all arcs incident on each node of V2. Hence, all SNCs in Kσ1,σ2 are V1,V2, and V1∪V2. Clearly, WncLμV1∪V2 is greater than WncLμV1 and WncLμV2. Hence,
(18)αs0LμKσ1,σ2=minWncLμV1,WncLμV2.
Similarly,αs0UμKσ1,σ2=minWncUμV1,WncUμV2. Also, WncLνV1∪V2 is less than WncLνV1 and WncLνV2. So,
(19)αs0LνKσ1,σ2=maxWncLνV1,WncLνV2.
In the same way, we haveαs0UνKσ1,σ2=maxWncUνV1,WncUνV2.Definition 9.
In an IVIFGG, two nodes are said to be SI if there is no SA between them. A set of nodes in G is an SI if and only if two nodes are in an SI set.Definition 10.
The weight of an SISD in an IVIFG G is described as
(20)WisD=WisLμD,WisUμD,WisLνD,WisUνD,i.e.,
(21)WisD=∑x∈DμBLxy,∑x∈DμBUxy,∑x∈DνBLxy,∑x∈DνBUxy,where μBLxy and μBUxy are minimum of the lower and upper of IVMBs and νBLxy and νBUxy are maximum of the lower and upper of IVNMBs of all SAs incident on x, respectively.
An SIN of an IVIFGG is shown by βs0G=βs0=βs0Lμ,βs0Uμ,βs0Lν,βs0Uν, which
(22)βs0Lμ=maxWisLμDDistheSISsofnodesinG,βs0Uμ=maxWisUμDDistheSISsofnodesinG,βs0Lν=minWisLνDDistheSISsofnodesinG,βs0Uν=minWisUνDDistheSISsofnodesinG.
A maximum SIS in an IVIFGG is an SIS with the maximum IVMBs and minimum IVNMBs.Theorem 11.
LetG be a CIVIFG. Then,
(23)βs0G=μBLxy,μBUxy,νBLxy,νBUxy,where μBLxy and μBUxy are the lower and upper of IVMBs and νBLxy and νBUxy are the lower and upper of IVNMBs of a weakest arc in G.Proof.
SinceG is a CIVIFG, so all arcs are strong, and also, each arc is neighbor to all other nodes. Hence, D=x is the only SIS for each x∈V. Thus, the result is true.Theorem 12.
LetKσ1,σ2 be a CB-IVIFG with partite set V1 and V2. Then,
(24)βs0LμKσ1,σ2=maxWisLμV1,WisLμV2,βs0UμKσ1,σ2=maxWisUμV1,WisUμV2,βs0LνKσ1,σ2=minWisLνV1,WisLνV2,βs0UνKσ1,σ2=minWisUνV1,WisUνV2.Proof.
InKσ1,σ2 all arcs are strong. Also, each node in V1 is neighbor with all nodes in V2 and contrariwise. Therefore, all SISs in Kσ1,σ2 are V1 and V2. Hence, the result is true.Example 1.
Consider an IVIFGG is drawn in Figure 1.
Clearly, all arcs are strong, and all SNCs ofG are as follows:
(25)D1=y,t,D2=x,y,z,D3=x,z,t,D4=y,z,t,D5=x,y,t,D6=x,y,z,t.Figure 1
An IVIFGG for a strong node cover (SNC).Table2 shows the method of calculating the weight of SISs.Table 2
Calculating the weight of SISs.
DWncLμDWncUμDWncLνDWncUνDWncDy,t0.1+0.10.2+0.20.2+0.20.5+0.50.2,0.4,0.4,1x,y,z0.1+0.1+0.20.2+0.2+0.40.2+0.2+0.20.5+0.5+0.50.4,0.8,0.6,1.5x,z,t0.1+0.1+0.20.2+0.2+0.40.2+0.2+0.20.5+0.5+0.50.4,0.8,0.6,1.5y,z,t0.1+0.2+0.10.2+0.4+0.20.2+0.2+0.20.5+0.5+0.50.4,0.8,0.6,1.5x,y,t0.1+0.1+0.10.2+0.2+0.20.2+0.2+0.20.5+0.5+0.50.3,0.6,0.6,1.5x,y,z,t0.1+0.1+0.2+0.10.2+0.2+0.4+0.20.2+0.2+0.2+0.20.5+0.5+0.5+0.50.5,1,0.8,2Thus,αs0=0.2,0.4,0.8,2.Example 2.
Consider a strong IVIFGG is drawn in Figure 2. All SISs in G are D1=x,z,D2=y,t. The calculation of the weight of SIS is shown in Table 3. Therefore, βs0=0.4,1,0.7,0.9.Definition 13.
LetG be an IVIFG without INs. The weight of an SAC Y is described as WacY=WacLμY,WacUμY,WacLνY,WacUνY, which WacY=∑xy∈YμBLxy,∑xy∈YμBUxy,∑xy∈YνBLxy,∑xy∈YνBUxy.
An SACN of an IVIFGG is denoted by αs1G=αs1=αs1Lμ,αs1Uμ,αs1Lν,αs1Uν, where
(26)αs1Lμ=minWacLμYYistheSACsofG,αs1Uμ=minWacUμYYistheSACsofG,αs1Lν=maxWacLνYYistheSACsofG,αs1Uν=maxWacUνYYistheSACsofG.
A minimum SAC in an IVIFGG is an SAC with minimum IVMBs and maximum IVNMBs.Theorem 14.
IfG is a complete IVIFG, then
(27)αs1Lμ=minWacLμYYisaSACinGwithY≥n2,αs1Uμ=minWacUμYYisaSACinGwithY≥n2,αs1Lν=maxWacLνYYisaSACinGwithY≥n2,αs1Uν=maxWacUνYYisaSACinGwithY≥n2.Proof.
SinceG is a CIVIFG, so all arcs are SA, and each vertex is neighbor to all others vertices. Also, the number of arcs in SAC of both G and G∗ is identical because each arc in both graphs is strong. Now, the SACN of G∗ is n/2. Therefore, the minimum number of arcs in an SAC of G is n/2. This completes the proof.Theorem 15.
IfKσ1,σ2 is a CB-IVIFG with partite set V1 and V2. Then,
(28)αs1LμKσ1,σ2=minWacLμYYisaSACinKσ1,σ2withY≥maxV1,V2,αs1UμKσ1,σ2=minWacUμYYisaSACinKσ1,σ2withY≥maxV1,V2,αs1LνKσ1,σ2=maxWacLνYYisaSACinKσ1,σ2withY≥maxV1,V2,αs1UνKσ1,σ2=maxWacUνYYisaSACinKσ1,σ2withY≥maxV1,V2.Proof.
InKσ1,σ2, all arcs are strong. Also, each node in V1 is neighbor with all nodes in V2 and contrariwise. Also, the number of arcs in an SAC of both G and G∗ is identical because each arc in both graph is strong. Now, the arc covering number of Kσ1,σ2∗ is maxV1,V2. Therefore, the minimum number of arcs in an SAC of Kσ1,σ2 is maxV1,V2. Thus, the result is obtained.Definition 16.
LetG be an IVIFG. A set T of SAs in G so that no two arcs in T have a common node is named an SIS of arcs or an SM in G.Definition 17.
LetT be an SM in IVIFG G. If xy∈T, then, we say that T strongly matches x to y. The weight of an SM is described as
(29)WsmT=WsmLμT,WsmUμT,WsmLνT,WsmUνT,WsmT=∑xy∈TμBLxy,∑xy∈TμBUxy,∑xy∈TνBLxy,∑xy∈TνBUxy.
An SMN of an IVIFGG is shown by βs1G=βs1=βs1Lμ,βs1Uμ,βs1Lν,βs1Uν, which
(30)βs1Lμ=maxWsmLμTTistheSMofG,βs1Uμ=maxWsmUμTTistheSMofG,βs1Lν=minWsmLνTTistheSMofG,βs1Uν=minWsmUνTTistheSMofG.
A maximum SM in an IVIFGG is an SM of maximum IVMBs and minimum IVNMBs.Theorem 18.
IfG is a CIVIFG, then
(31)βs1Lμ=maxWsmLμTTisaSMwithT≤n2,βs1Uμ=maxWsmUμTTisaSMwithT≤n2,βs1Lν=minWsmLνTTisaSMwithT≤n2,βs1Uν=minWsmUνTTisaSMwithT≤n2.Proof.
SinceG is a CIVIFG, all arcs are strong, and each node is neighbor to all other nodes. Also, the number of arcs in an SM of both G and G∗ is identical because each arc in both graph is strong. Now, the SMN of G∗ is n/2. Therefore, the maximum number of arcs in an SM of G is n/2. Hence, the result follows.Theorem 19.
For a CB-IVIFGKσ1,σ2 with partite set V1 and V2,
(32)βs1LμKσ1,σ2=maxWsmLμTTisaSMinKσ1,σ2withT≤minV1,V2,βs1UμKσ1,σ2=maxWsmUμTTisaSMinKσ1,σ2withT≤minV1,V2,βs1LνKσ1,σ2=minWsmLνTTisaSMinKσ1,σ2withT≤minV1,V2,βs1UνKσ1,σ2=minWsmUνTTisaSMinKσ1,σ2withT≤minV1,V2.Proof.
InKσ1,σ2, all arcs are strong. Also, each node in V1 is neighbor with all nodes in V2 and contrariwise. Thus, the number of arcs in an SM of both Kσ1,σ2 and Kσ1,σ2∗ is identical because each arc in both graphs is strong. Now, the SMN of Kσ1,σ2∗ is maxV1,V2. Therefore, the maximum number of arcs in an SM of Kσ1,σ2 is maxV1,V2.
Hence, the result is obtained.Example 3.
Consider a strong IVIFGG is drawn in Figure 3. All arcs are strong, and the SACs are as follows:
(33)Y1=xy,tz,Y2=xt,yz,Y3=yt,tx,yz,Y4=yt,xy,tz,Y5=xy,yz,zt,Y6=xy,xt,tz,Y7=xt,xy,yz,Y8=xt,zt,yz.Figure 2
A strong IVIFGG for a strong independent set (SIS).Table 3
Calculating the weight of SISs.
DWncLμDWncUμDWncLνDWncUνDWncDx,t0.2+0.20.5+0.40.4+0.40.5+0.50.4,0.9,0.8,1y,t0.2+0.20.5+0.50.3+0.40.4+0.50.4,1,0.7,0.9Figure 3
A strong IVIFGG for strong maching (SM).The calculation of the weight of SACs is shown in Table4.Table 4
Calculating the weight of strong arc cover setsY.
YWacLμYWacUμYWacLνYWacUνYWacYY10.1+0.20.2+0.40.2+0.10.5+0.30.3,0.6,0.3,0.8Y20.1+0.20.2+0.50.2+0.20.4+0.50.3,0.7,0.4,0.9Y30.2+0.1+0.20.4+0.2+0.50.2+0.2+0.20.5+0.4+0.50.5,1.1,0.6,1.4Y40.2+0.1+0.20.4+0.2+0.40.2+0.2+0.10.5+0.5+0.30.5,1,0.5,1.3Y50.1+0.2+0.20.2+0.5+0.40.2+0.2+0.10.5+0.5+0.30.5,1.1,0.5,1.3Y60.1+0.1+0.20.2+0.2+0.40.2+0.2+0.10.5+0.4+0.30.4,0.8,0.5,1.2Y70.1+0.1+0.20.2+0.2+0.50.2+0.2+0.20.4+0.5+0.50.4,0.9,0.6,1.4Y80.1+0.2+0.20.2+0.4+0.50.2+0.1+0.20.4+0.3+0.50.4,1.1,0.5,1.2So,αs1=0.3,0.6,0.6,1.4.Again, the two setsY1 and Y2 are the only SAC and SM in G. So,
(34)WsmY1=0.3,0.6,0.3,0.8,WsmY2=0.3,0.7,0.4,0.9.Hence,βs1=0.3,0.7,0.3,0.8.Example 4.
Consider an IVIFGG is drawn in Figure 4.
All SAs arext, zt, and yz, and all SACs are as follows:
(35)Y1=xt,yz,Y2=xt,tz,yz.
The calculation of the weight of SACs is shown in Table5. So, αs1=0.4,1,1.1,1.4.Figure 4
An IVIFGG for strong independent arc cover (SIAC).Table 5
Calculating the weight of strong arc cover setsY.
YWacLμYWacUμYWacLνYWacUνYWacYY10.2+0.20.5+0.50.4+0.30.5+0.40.4,1,0.7,0.9Y20.2+0.2+0.20.5+0.5+0.50.4+0.4+0.30.5+0.5+0.40.6,1.5,1.1,1.4The setY1=xt,yz is the only SIAC. So, βs1=WsmY1=0.4,1,0.7,0.9.Theorem 20.
LetG be an IVIFG containing no IN. Then,
(36)αs0Lμ+βs0Lμ=WLμV,αs0Lν+βs0Lν=WLνV,αs0Uμ+βs0Uμ=WUμV,αs0Uν+βs0Uν=WUνV.Proof.
LetMs0 be a minimum SNC of G, which
(37)αs0Lμ=WLμMs0,αs0Lν=WLνMs0,αs0Uμ=WUμMs0,αs0Uν=WUνMs0.
Then,V−Ms0 is an SIS of nodes. In other words, the nodes in V−Ms0 are incident on SAs of G. Thus,
(38)βs0Lμ≥WLμV−Ms0=WLμV−αs0Lμ⇒αs0Lμ+βs0Lμ≥WLμV,βs0Uμ≥WUμV−Ms0=WUμV−αs0Uμ⇒αs0Uμ+βs0Uμ≥WUμV,βs0Lν≤WLνV−Ms0=WLνV−αs0Lν⇒αs0Lν+βs0Lν≤WLνV,βs0Uν≤WUνV−Ms0=WUνV−αs0Uν⇒αs0Uν+βs0Uν≤WUνV.
Letβs0Lμ=WQs0, where Qs0 is a maximum SIS of nodes in G. That is, no two nodes in Qs0 are neighbor to each other by an SA, and thus, the node in V−Qs0 strongly covers all SAs of G. Hence, V−Qs0 is an SNC of G, and αs0Lμ and αs0Uμ are the minimum lower and upper of IVMBs, and αs0Lν and αs0Uν are the maximum lower and upper of IVNMBs. So,
(39)αs0Lμ≤WLμV−Qs0=WLμV−βs0Lμ⇒αs0Lμ+βs0Lμ≤WLμV,αs0Uμ≤WUμV−Qs0=WUμV−βs0Uμ⇒αs0Uμ+βs0Uμ≤WUμV,αs0Lν≥WLνV−Qs0=WLνV−βs0Lν⇒αs0Lν+βs0Lν≥WLνV,αs0Uν≥WUνV−Qs0=WUνV−βs0Uν⇒αs0Uν+βs0Uν≥WUνV.
From (38) and (39), we have
(40)αs0Lμ+βs0Lμ=WLμV,αs0Uμ+βs0Uμ=WUμV,αs0Lν+βs0Lν=WLνV,αs0Uν+βs0Uν=WUνV.Definition 21.
LetG be an IVIFG and M be an SM in G. Then, M is named a PSM if M strongly matches each node of G to some nodes of G.Example 5.
Consider an IVIFGG is drawn in Figure 5. All arcs are strong, and the sets M1 and M2 are PSMs. The calculation of the weight of PSMs is given in Table 6.
(41)M1=xt,yz,M2=xz,yt,M3=xt,xz,yz,M4=xt,ty,yz.
So,βs1=0.6,1.3,0.7,0.9.
Hence,αs1=0.3,0.7,1.2,1.6.Figure 5
An IVIFGG for perfect strong matching (PSM).Table 6
Calculating the weight of perfect strong matchingsM.
MWsmLμMWsmUμMWsmLνMWsmUνMWsmMM10.3+0.10.6+0.20.2+0.50.4+0.60.4,0.8,0.7,1M20.1+0.20.2+0.50.5+0.20.6+0.30.3,0.7,0.7,0.9M30.3+0.1+0.10.6+0.2+0.20.2+0.5+0.50.4+0.6+0.60.5,1,1.2,1.6M40.3+0.2+0.10.6+0.5+0.20.2+0.2+0.50.4+0.3+0.60.6,1.3,0.9,1.3Now, we introduced PD in IVIFGs using SAs based on PSM. Also, some useful results are established.Definition 22.
A setD of nodes of IVIFG G is an SDS of G if every node of V−D is a strong neighbor of some nodes in D.Definition 23.
The weight of an SDSD is defined as
(42)WsdD=WsdLμD,WsdUμD,WsdLνD,WsdUνD,or
(43)WsdD=∑x∈DμBLxy,∑x∈DμBUxy,∑x∈DνBLxy,∑x∈DνBUxy,where μBLxy and μBUxy are the minimum lower and upper of IVMBs and νBLxy and νBUxy are the maximum lower and upper of IVNMBs of SAs incident on x, respectively.
An SDN of an IVIFGG is denoted by γsG=γs=γsLμ,γsUμ,γsLν,γsUν, where
(44)γsLμ=minWsdLμDDistheSDSsofG,γsUμ=minWsdUμDDistheSDSsofG,γsLν=maxWsdLνDDistheSDSsofG,γsUν=maxWsdUνDDistheSDSsofG.Definition 24.
LetG be an IVIFG. A set D of nodes is named to be an SPDS if D is an SDS and the IVIF-subgraph induced by D has a PSM. The weight of an SPDS D is described as WspdD=WspdLμD,WspdUμD,WspdLνD,WspdUνD, which
(45)WspdD=∑x∈DμBLxy,∑x∈DμBUxy,∑x∈DνBLxy,∑x∈DνBUxy.
An SPDN of an IVIFGG is denoted by γspdG=γspd=γspdLμ,γspdUμ,γspdLν,γspdUν, that
(46)γspdLμ=minWspdLμDDistheSPDSsofG,γspdUμ=minWspdUμDDistheSPDSsofG,γspdLν=maxWspdLνDDistheSPDSsofG,γspdUν=maxWspdUνDDistheSPDSsofG.Example 6.
Consider an IVIFGG is drawn in Figure 6. All SAs are xt,zt, and yz. The PDs in G are D1, D2, and D3. The weights of these sets are calculated as follows:
(47)D2=x,y,D2=z,t,D3=x,y,z,t.Figure 6
An IVIFGG for strong paired dominating set (SPDS).Table7 shows the calculation of the weight of PDs.Table 7
Calculating the weight of paired dominating sets.
DWspdLμDWspdUμDWspdLνDWspdUνDWspdDD10.2+0.20.5+0.50.4+0.20.5+0.40.4,1,0.6,0.9D20.2+0.20.5+0.50.4+0.40.5+0.50.4,1,0.8,1D30.2+0.2+0.2+0.20.5+0.5+0.5+0.50.4+0.2+0.4+0.40.5+0.4+0.5+0.50.8,2,1.4,1.9Hence,γspdG=0.4,1,1.4,1.9.Theorem 25.
LetG be a CIVIFG. Then,
(48)γspdLμ=2μBLxy,γspdLν=2νBLxy,γspdUμ=2μBUxy,γspdUν=2νBUxy,where μBLxy and μBUxy are the lower and upper of IVMB and νBLxy and νBUxy are the lower and upper of IVNMB of any weakest arc in G, respectively.Proof.
SinceG is a CIVIFG, all arcs are strong, and every vertex is neighbor to all other vertices. Then, any set consisting of two nodes x1,x2 in G forms an SPDS. Hence,
(49)γspdLμ=μBLxy+μBLxy=2μBLxy,γspdUμ=μBUxy+μBUxy=2μBUxy,γspdLν=νBLxy+νBLxy=2νBLxy,γspdUν=νBUxy+νBUxy=2νBUxy,where xy is the weakest arc in G.


## 4. Application

Social networks are a group of individuals or organizations with common tastes or interests that come together to achieve specific goals. Each member is named an actor. Social networks are characterized by complex relationships and interactions between actors. The main reasons for creating social networks are individual relationships, labor relations, scientific relations, shared tastes, interests and hobbies, sociopolitical motives, and virtual network analysis.Graphs are used as a mathematical tool to represent and analyze a social network by visually representing social networks. In these graphs, the actors are considered as vertices of the graph, and the connections between them are displayed by the edges of the graph. Intuitively, the edges are distributed on social networks locally. This means that the number of edges distributed among a group of vertices is much greater than the number of distribution edges among this group of vertices and the rest of the vertices of the graph. This feature, which can be seen in graphs related to real data, is called a community. In some sources, the community is also called a cluster or module. In other words, communities are a set of vertices that are more likely to share common features than the rest of the graph. Since people in forums on a social network are more likely to have common interests, this information can be used to promote specific products by finding their interests. Most online social networks have overlapping communities. This means that these networks are made up of overlapping communities, and one vertex can belong to more than one community. Figure7 illustrates the social network of researchers in a country that is a member of different scientific communities according to the subjects under study. These communities include chemistry, biology, engineering, information technology (IT), mathematics, medicine, physics, and social sciences. Table 8 shows the number of members of each community and the average number of members present at the meetings.Figure 7
Scientific community network of researchers.Table 8
Data set.
VerticesCommunitiesNumber of membersAverage attendance of membersaBiology3530bChemistry2521cEngineering5545dInformation technology (IT)7560eMathematics4032fMedicine5045gPhysics3024hSocial sciences4537In the evaluations made by the members on the effect of community on the scientific promotion of members, since the mentioned variables have uncertain values, so for each community, we considered an interval-valued intuitionistic fuzzy number as the amount of influence of community on its members. Since the presence of members in the meetings of the community is effective on the scientific promotion of members, we introduced the ratio of the average number of members present in the meetings to the total number as an IVIFN. For example, studies have shown that the biology community is 80 to 90 percent effective in advancing the science of its members and 5 to 10 percent ineffective. These values are specified in Table9.Table 9
The IVIFNs of scientific communities.
VerticesCommunitiesThe IVIFNs corresponding to each community.aBiology0.80,0.90,0.05,0.10bChemistry0.79,0.89,0.11,0.21cEngineering0.76,0.86,0.14,0.16dInformation technology (IT)0.75,0.85,0.10,0.15eMathematics0.75,0.85,0.10,0.15fMedicine0.80,0.90,0.5,0.10gPhysics0.75,0.85,0.10,0.15hSocial sciences0.77,0.87,0.10,0.13The strong relationships between scientific communities are illustrated in the form of an IVIFG in Figure8. In this IVIFG, the membership values of the edges are the effect that the members of the two communities have on their scientific advancement. For example, the collaboration between the two communities of chemistry and medicine is about 79 to 89 percent effective in the scientific advancement of the members of each community and 11 to 21 percent ineffective. These values are shown in Table 10.Figure 8
The scientific communities IVIFG.Table 10
IVIFNs of relations between scientific communities.
EdgesIVIFNsEdgesIVIFNsab0.97,0.89,0.11,0.21df0.75,0.85,0.10,0.15ac0.76,0.86,0.14,0.16dg0.75,0.85,0.10,0.15ad0.75,0.85,0.10,0.15dh0.75,0.85,0.10,0.15be0.75,0.85,0.11,0.21ef0.75,0.85,0.10,0.15bf0.79,0.89,0.11,0.21eh0.75,0.85,0.10,0.15cd0.75,0.85,0.14,0.16fh0.77,0.87,0.10,0.13cf0.76,0.86,0.14,0.16gh0.75,0.85,0.10,0.15ce0.75,0.85,0.14,0.16cg0.75,0.85,0.14,0.16In general, there is no polynomial algorithm for finding a maximum independent set for an arbitrary graph. This means that it is not possible to access such a collection in a short time. To obtain the maximal SISs in IVIFG with a small number of vertices, we use the following instructions.Algorithm 1: Finding the maximal SISs F of G containing an arbitrary vertex x.
Step 1. Consider vertex x as a member of F. Then, remove all adjacent vertices of x.Step 2. Consider another arbitrary vertex in the remaining graph as a new member ofF.Depending on which member of the remaining vertex set is selected, different independent sets, includingx, are obtained.Step 3. Repeat Step 2 to select all possible vertices.Since all edges are SA, so by applying the above steps for all vertices on the IVIFG of Figure8, all maximal SISs and cardinalities can be seen in Table 11. Now, by calculating the cardinal of all the SISs obtained from the above steps, we can also determine the maximum SISs.Table 11
Calculations for finding maximal SISs in the IVIFG of Figure8.
Step 1Step 2Step 3SISsaeg{a,e,g}fg{a, f, g}gf{a,g,f}e{a,g,e}h{a,h}bch{b,c,h}d{b,d}g{b,g}hc{b,h,c}cbh{c,b,h}hb{c,h,b}db{d,b}e{d,e}eag{e,a,g}d{e,d}ga{e,g,a}fag{f,a,g}ga{f,g,e}gaf{g,a,f}e{g,a,e}b{g,b}ea{g,e,a}fa{g,f,a}ha{h,a}bc{h, b, c}cb{h,c,b}The maximum SISs areD1=a,e,g, D2=a,f,g, and D3=b,c,h.After calculating the weight of the above sets, we haveWD1=2.25,2.55,0.42,0.58, WD2=2.25,2.55,0.42,0.53, and WD3=2.25,2.55,0.35,0.52.Therefore,D3 has the maximum weight of membership and the minimum weight of nonmembership, so it can be chosen as the best option. It is interesting to know that D3 also has the maximum number of members in scientific communities. That is, strong independent scientific communities include chemistry, engineering, and social sciences.Suppose knowledge-based companies intend to organize an exhibition at the meeting place of scientific communities to acquaint researchers with their scientific products. Researchers at the knowledge-based companies can be members of various scientific communities. The goal is to hold as many exhibitions as possible at the same time provided that each knowledge-based company has a maximum of one exhibition in a specific time period and to hold another exhibition at different time intervals. In this case, the maximum independent set is the maximum number of exhibitions that can be held at one time in scientific communities.


## 5. Conclusion

Analysis of uncertain problems by IVIFG is important because it gives more integrity and flexibility to the system. An IVIFG, as an extension of FGs, has good capabilities in dealing with problems that cannot be explained by FGs. They have been able to have wide applications even in fields such as psychology and identifying people based on cancerous behaviors. In this paper, covering and matching have been defined in IVIFGs using strong arcs. These concepts are introduced as an interval-valued intuitionistic number. One of the advantages of this method is that the amount of defined parameters can be expressed and compared in terms of membership and nonmembership. Also, the concepts of SNC, SIN, SAC, and SM in IVIFGs are determined, and the relations among them have been obtained. Furthermore, we have introduced the PD and SPDN in CIVIFG and CB-IVIFG. Since the parameters being studied are interval values, comparisons of these parameters may be limited in an IVIFG. Finally, we have presented an application of IVIFG in social networks. In their future work, the authors try to study the concepts ofm-polar IVIFGs.



---

*Source: 2412012-2022-07-08.xml*