Neutrosophic Soft Rough Graphs with Application

axioms Article

Neutrosophic Soft Rough Graphs with Application Muhammad Akram 1, * 1 2

*

ID

, Hafsa M. Malik 1 , Sundas Shahzadi 1 and Florentin Smarandache 2

ID

Department of Mathematics, University of the Punjab, New Campus, Lahore 54590, Pakistan; hafsa.masood.malik@gmail.com (H.M.M.); sundas1011@gmail.com (S.S.) Mathematics & Science Department, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA; fsmarandache@gmail.com Correspondence: m.akram@pucit.edu.pk; Tel.: +92-42-99231241

Received: 27 January 2018; Accepted: 23 February 2018; Published: 26 February 2018

Abstract: Neutrosophic sets (NSs) handle uncertain information while fuzzy sets (FSs) and intuitionistic fuzzy sets (IFs) fail to handle indeterminate information. Soft set theory, neutrosophic set theory, and rough set theory are different mathematical models for handling uncertainties and they are mutually related. The neutrosophic soft rough set (NSRS) model is a hybrid model by combining neutrosophic soft sets with rough sets. We apply neutrosophic soft rough sets to graphs.

In this research paper, we introduce the idea of neutrosophic soft rough graphs (NSRGs) and describe different methods of their construction. We consider the application of NSRG in decision-making problems. In particular, we develop efficient algorithms to solve decision-making problems.

Keywords: neutrosophic soft rough sets; neutrosophic soft rough graphs; decision-making; algorithm

1. Introduction Smarandache [1] initiated the concept of neutrosophic set (NS). Smarandache’s NS is characterized by three parts: truth, indeterminacy, and falsity. Truth, indeterminacy and falsity membership values behave independently and deal with problems having uncertain, indeterminant and imprecise data. Wang et al. [2] gave a new concept of single valued neutrosophic sets (SVNSs) and defined the set theoretic operators on an instance of NS called SVNS. Peng et al. [3] discussed the operations of simplified neutrosophic numbers and introduced an outranking idea of simplified neutrosophic numbers.

Molodtsov [4] introduced the notion of soft set (SS) as a novel mathematical approach for handling uncertainties. Molodtsov’s SSs gave us a new technique for dealing with uncertainty from the viewpoint of parameters. Maji et al. [5–7] introduced neutrosophic soft sets (NSSs), intuitionistic fuzzy soft sets and fuzzy soft sets (FSSs). In [8], Sahin and Kucuk presented NSS in the form of neutrosophic relations.

Theory of rough set (RS) was proposed by Pawlak [9] in 1982. Rough set theory is used to study the intelligence systems containing incomplete, uncertain or inexact information. The lower and upper approximation operators of RSs are used for managing hidden information in a system. Feng et al. [10] took a significant step to introduce parametrization tools in RSs. Meng et al. [11] provide further discussion of the combination of SSs, RSs and FSs. The existing results of RSs and other extended RSs such as rough fuzzy sets, generalized rough fuzzy sets, soft fuzzy rough sets and intuitionistic fuzzy rough sets based decision-making models have their advantages and limitations [12,13].

In a different way, rough set approximations have been constructed into the intuitionistic fuzzy environment and are known as intuitionistic fuzzy rough sets and rough intuitionistic fuzzy sets [14,15].

Zhang et al. [16,17] presented the notions of soft rough sets, soft rough intuitionistic fuzzy sets, intuitionistic fuzzy soft rough sets, its application in decision-making, and also introduced generalized intuitionistic fuzzy soft rough sets. Broumi et al. [18,19] developed a hybrid structure by combining Axioms 2018, 7, 14; doi:10.3390/axioms7010014

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RSs and NSs, called RNSs, they also presented interval valued neutrosophic soft rough sets by combining interval valued neutrosophic soft sets and RSs. Yang et al. [20] proposed single valued neutrosophic rough sets (SVNRSs) by combining SVNSs and RSs and defined SVNRSs on two universes and established an algorithm for a decision-making problem based on SVNRSs on two universes.

Akram and Nawaz [21] have introduced the concept of soft graphs and some operation on soft graphs. Certain concepts of fuzzy soft graphs and intuitionistic fuzzy soft graphs are discussed in [22–24]. Akram and Shahzadi [25] have introduced neutrosophic soft graphs. Zafar and Akram [26] introduced a rough fuzzy digraph and several basic notions concerning rough fuzzy digraphs. In this research paper, a neutrosophic soft rough set is a generalization of a neutrosophic rough set, and we introduce the idea of neutrosophic soft rough graphs (NSRGs) that are made by combining NSRSs with graphs and describe different methods of their construction. We consider the application of NSRG in decision-making problems and resolve the problem. In particular, we develop efficient algorithms to solve decision-making problems.

For other notations, terminologies and applications not mentioned in the paper, the readers are referred to [27–35].

2. Neutrosophic Soft Rough Information In this section, we will introduce the notions of neutrosophic soft rough relation (NSRR), and NSRGs. Definition 1. Let Y be an initial universal set, P a universal set of parameters and M Ď P. For an arbitrary neutrosophic soft relation Q over Y ˆ M, pY, M, Qq is called neutrosophic soft approximation space (NSAS).

For any NS A P N pMq, we define the upper neutrosophic soft rough approximation (UNSRA) and the lower neutrosophic soft rough approximation (LNSRA) operators of A with respect to pY, M, Qq denoted by QpAq and QpAq, respectively as follows: QpAq

“

tpu, TQp Aq puq, IQp Aq puq, FQp Aq puqq | u P Yu,

QpAq

“

tpu, TQp Aq puq, IQp Aq puq, FQp Aq puqq | u P Yu,

where TQp Aq puq

“

ł`

˘ TQp Aq pu, eq ^ TA peq ,

e PM

FQp Aq puq

“

IQp Aq puq

“

ľ`

˘

FQp Aq pu, eq _ FA peq ;

IQp Aq puq “ TQp Aq puq “

˘ IQp Aq pu, eq _ I A peq ,

ePM

ľ`

˘ FQp Aq pu, eq _ TA peq ,

ePM

e PM

ł`

ľ`

ł` ˘ ˘ p1 ´ IQp Aq pu, eqq ^ I A peq , FQp Aq puq “ TQp Aq pu, eq ^ FA peq .

e PM

e PM

The pair pQpAq, QpAqq is called NSRS of A w.r.t pY, M, Qq, Q and Q are referred to as the LNSRA and the UNSRA operators, respectively.

Example 1. Suppose that Y “ tw1 , w2 , w3 , w4 u is the set of careers under consideration, and Mr. X wants to select the best suitable career. M “ te1 , e2 , e3 u is a set of decision parameters. Mr. X describes the “most suitable career" by defining a neutrosophic soft set pQ, Mq on Y that is a neutrosophic relation from Y to M as shown in Table 1.

Table 1. Neutrosophic soft relation Q. Q

w1

w2

w3

w4

e1 e2 e3

p0.3, 0.4, 0.5q p0.1, 0.5, 0.4q p0.3, 0.4, 0.4q

p0.4, 0.2, 0.3q p0.3, 0.4, 0.6q p0.4, 0.6, 0.7q

p0.1, 0.5, 0.4q p0.4, 0.4, 0.3q p0.3, 0.5, 0.4q

p0.2, 0.3, 0.4q p0.5, 0.3, 0.8q p0.5, 0.4, 0.6q

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Now, Mr. X gives the most favorable decision object A, which is an NS on M defined as follows: A “ tpe1 , 0.5, 0.2, 0.4q, pe2 , 0.2, 0.3, 0.1q, pe3 , 0.2, 0.4, 0.6qu. By Definition 1, we have TQp Aq pw1 q “ 0.3, IQp Aq pw1 q “ 0.4, FQp Aq pw1 q “ 0.4, TQp Aq pw2 q “ 0.4, IQp Aq pw2 q “ 0.2, FQp Aq pw2 q “ 0.4, TQp Aq pw3 q “ 0.2, IQp Aq pw3 q “ 0.4, FQp Aq pw3 q “ 0.3, TQp Aq pw4 q “ 0.2, IQp Aq pw4 q “ 0.3, FQp Aq pw4 q “ 0.4.

Similarly, TQp Aq pw1 q “ 0.4, IQp Aq pw1 q “ 0.4, FQp Aq pw1 q “ 0.3, TQp Aq pw2 q “ 0.5, IQp Aq pw2 q “ 0.4, FQp Aq pw2 q “ 0.4, TQp Aq pw3 q “ 0.4, IQp Aq pw3 q “ 0.4, FQp Aq pw3 q “ 0.3, TQp Aq pw4 q “ 0.5, IQp Aq pw4 q “ 0.4, FQp Aq pw4 q “ 0.5.

Thus, we obtain QpAq

“

tpw1 , 0.3, 0.4, 0.4q, pw2 , 0.4, 0.2, 0.4q, pw3 , 0.2, 0.4, 0.3q, pw4 , 0.2, 0.3, 0.4qu,

QpAq

“

tpw1 , 0.4, 0.4, 0.3q, pw2 , 0.5, 0.4, 0.4q, pw3 , 0.4, 0.4, 0.3q, pw4 , 0.5, 0.4, 0.5qu.

Hence, pQpAq, QpAqq is an NSRS of A. The conventional neutrosophic soft set is a mapping from a parameter to the neutrosophic subset of the universe and letting pQ, Mq be neutrosophic soft set. Now, we present the constructive definition of neutrosophic soft rough relation by using a neutrosphic soft relation S from M ˆ M “ Ḿ to N pY ˆ Y “ Ýq, where Y is a universal set and M be a set of parameters.

Definition 2. A neutrosophic soft rough relation pSpBq, SpBqq on Y is an NSRS, S : Ḿ Ñ N pÝq is a neutrosophic soft relation on Y defined by Spei e j q

“

tui u j | Dui P Qpei q, u j P Qpe j qu, ui u j P Ý, such that

TS pui u j , ei e j q

ď

mintTQ pui , ei q, TQ pu j , e j qu

IS pui u j , ei e j q

ď

maxtIQ pui , ei q, IQ pu j , e j qu

FS pui u j , ei e j q

ď

For any B P N pḾq, B

“

maxtFQ pui , ei q, FQ pu j , e j qu. ` ˘ t ei e j , TB pei e j q, IB pei e j q, FB pei e j q ui u j P Ḿu,

TB pei e j q

ď

mintTA pei q, TA pe j qu,

IB pei e j q

ď

maxtI A pei q, I A pe j qu,

FB pei e j q

ď

maxtFA pei q, FA pe j qu.

The UNSA and the LNSA of B w.r.t pÝ, Ḿ, Sq are defined as follows: SpBq “ tpui u j , TSpBq pui u j q, ISpBq pui u j q, FSpBq pui u j qq | ui u j P Ýu, SpBq “ tpui u j , TSpBq pui u j q, ISpBq pui u j q, FSpBq pui u j qq | ui u j P Ýu,

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where TSpBq pui u j q

ł `

“

˘ TS pui u j , ei e j q ^ TB pei e j q ,

ei e j PḾ

ISpBq pui u j q

ľ `

“

˘ IS pui u j , ei e j q _ IB pei e j q ,

ei e j PM̃

FSpBq pui u j q

ľ `

“

˘ FS pui u j , ei e j q _ FB pei e j q ;

ei e j PM̃

TSpBq pui u j q

“

ľ `

˘ FS pui u j , ei e j q _ TB pei e j q ,

ei e j PḾ

ISpBq pui u j q

“

ł `

˘ p1 ´ IS pui u j , ei e j qq ^ IB pei e j q ,

ei e j PM̃

FSpBq pui u j q

“

ł `

˘ TS pui u j , ei e j q ^ FB pei e j q .

ei e j PM̃

The pair pSpBq, SpBqq is called NSRR and S, S : N pḾq Ñ N pÝq are called the LNSRA and the UNSRA operators, respectively.

Remark 1. Consider an NS B on Ḿ and an NS A on M, according to the definition of NSRR, we get TSpBq pui u j q ISpBq pui u j q

ď mintTSp Aq pui q, TSp Aq pu j qu, ď maxtISp Aq pui q, ISp Aq pu j qu,

FSpBq pui u j q

ď maxtFSp Aq pui q.FSp Aq pu j qu.

Similarly, for LNSRA operator SpBq, TSpBq pui u j q ISpBq pui u j q

ď mintTSp Aq pui q, TSp Aq pu j qu, ď maxtISp Aq pui q, ISp Aq pu j qu,

FSpBq pui u j q

ď maxtFSp Aq pui q.FSp Aq pu j qu.

Example 2. Let Y “ tu1 , u2 , u3 u be a universal set and M “ te1 , e2 , e3 u a set of parameters. A neutrosophic soft set pQ, Mq on Y can be defined in tabular form in Table 2 as follows: Table 2. Neutrosophic soft set pQ, Mq. Q

u1

u2

u3

e1 e2 e3

p0.4, 0.5, 0.6q p0.5, 0.3, 0.6q p0.7, 0.2, 0.3q

p0.7, 0.3, 0.2q p0.3, 0.4, 0.3q p0.6, 0.5, 0.4q

p0.6, 0.3, 0.4q p0.7, 0.2, 0.3q p0.7, 0.2, 0.4q

Let E “ tu1 u2 , u2 u3 , u2 u2 , u3 u2 u Ď Ý and L “ te1 e3 , e2 e1 , e3 e2 u Ď Ḿ.

Then, a soft relation S on E (from L to E) can be defined in Table 3 as follows:

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Table 3. Neutrosophic soft relation S. S

u1 u2

u2 u3

u2 u2

u3 u2

e1 e3 e2 e1 e3 e2

p0.4, 0.4, 0.5q p0.3, 0.3, 0.4q p0.3, 0.3, 0.2q

p0.6, 0.3, 0.4q p0.3, 0.2, 0.3q p0.5, 0.3, 0.2q

p0.5, 0.4, 0.2q p0.2, 0.3, 0.3q p0.2, 0.4, 0.4q

p0.5, 0.4, 0.3q p0.7, 0.2, 0.2q p0.3, 0.4, 0.4q

Let A “ tpe1 , 0.2, 0.4, 0.6q, pe2 , 0.4, 0.5, 0.2q, pe3 , 0.1, 0.2, 0.4qu be an NS on M, then SpAq “ tpu1 , 0.4, 0.2, 0.4q, pu2 , 0.3, 0.4, 0.3q, pu3 , 0.4, 0.2, 0.3qu, SpAq “ tpu1 , 0.3, 0.5, 0.4q, pu2 , 0.2, 0.5, 0.6q, pu3 , 0.4, 0.5, 0.6qu.

Let B “ tpe1 e3 , 0.1, 0.3, 0.5q, pe2 e1 , 0.2, 0.4, 0.3q, pe3 e2 , 0.1, 0.2, 0.3qu be an NS on L, then SpBq “ tpu1 u2 , 0.2, 0.3, 0.3q, pu2 u3 , 0.2, 0.3, 0.3q, pu2 u2 , 0.2, 0.4, 0.3q, pu3 u2 , 0.2, 0.4, 0.3qu, SpBq “ tpu1 u2 , 0.2, 0.4, 0.4q, pu2 u3 , 0.2, 0.4, 0.5q, pu2 u2 , 0.3, 0.4, 0.5q, pu3 u2 , 0.2, 0.4, 0.5qu.

Hence, SpBq “ pSpBq, SpBqq is NSRR. Definition 3. A neutrosophic soft rough graph (NSRG) on a non-empty V is an 4-ordered tuple pV, M, QpAq, SpBqq such that (i) (ii) (iii) (vi) (v) (iv)

M is a set of parameters, Q is an arbitrary neutrosophic soft relation over V ˆ M, S is an arbitrary neutrosophic soft relation over V́ ˆ Ḿ, QpAq “ pQA, QAq is an NSRS of A, SpBq “ pSB, SBq is an NSRR on V́ Ă V ˆ V, G “ pQpAq, SpBqq is a neutrosophic soft rough graph, where G “ pQA, SBq and G “ pQA, SBq are lower neutrosophic approximate graph (LNAG) and upper neutrosophic approximate graph (UNAG), respectively of neutrosophic soft rough graph (NSRG) G “ pQpAq, SpBqq.

Example 3. Let V “ tv1 , v2 , v3 , v4 , v5 , v6 u be a vertex set and M “ te1 , e2 , e3 u a set of parameters. A neutrosophic soft relation over V ˆ M can be defined in tabular form in Table 4 as follows: Table 4. Neutrosophic soft relation Q. Q

v1

v2

v3

v4

v5

v6

e1 e2 e3

p0.4, 0.5, 0.6q p0.5, 0.4, 0.2q p0.5, 0.4, 0.1q

p0.7, 0.3, 0.5q p0.6, 0.4, 0.5q p0.6, 0.3, 0.2q

p0.6, 0.2, 0.3q p0.7, 0.3, 0.4q p0.5, 0.4, 0.3q

p0.4, 0.4, 0.2q p0.5, 0.3, 0.2q p0.6, 0.2, 0.3q

p0.5, 0.5, 0.6q p0.4, 0.5, 0.4q p0.5, 0.4, 0.4q

p0.4, 0.5, 0.6q p0.6, 0.5, 0.4q p0.7, 0.3, 0.5q

Let A “ tpe1 , 0.5, 0.4, 0.6q, pe2 , 0.7, 0.4, 0.5q, pe3 , 0.6, 0.2, 0.5qu be an NS on M, then SpAq

“

tpv1 , 0.5, 0.4, 0.5q, pv2 , 0.6, 0.3, 0.5q, pv3 , 0.7, 0.4, 0.5q, pv4 , 0.6, 0.2, 0.5q, pv5 , 0.5, 0.4, 0.5q, pv6 , 0.6, 0.3, 0.5qu,

SpAq

“

tpv1 , 0.6, 0.4, 0.5q, pv2 , 0.5, 0.4, 0.6q, pv3 , 0.5, 0.4, 0.6q, pv4 , 0.5, 0.4, 0.5q, pv5 , 0.6, 0.4, 0.5q, pv6 , 0.6, 0.4, 0.5qu.

Let E “ tv1 v1 , v1 v2 , v2 v1 , v2 v3 , v4 v5 , v3 v4 , v5 v2 , v5 v6 u Ď V́ and L “ te1 e3 , e2 e1 , e3 e2 u Ď Ḿ.

Then, a neutrosophic soft relation S on E (from L to E) can be defined in Tables 5 and 6 as follows: Table 5. Neutrosophic soft relation S. S

v1 v1

v1 v2

v2 v1

v2 v3

e1 e2 e2 e3 e1 e3

p0.4, 0.4, 0.2q p0.5, 0.4, 0.1q p0.4, 0.4, 0.1q

p0.4, 0.4, 0.5q p0.4, 0.3, 0.2q p0.4, 0.2, 0.2q

p0.4, 0.4, 0.5q p0.4, 0.3, 0.2q p0.4, 0.2, 0.2q

p0.6, 0.3, 0.4q p0.5, 0.3, 0.2q p0.5, 0.3, 0.3q

Table 5: Neutrosophic v1 v1 v1 v2 (0.4, 0.4, 0.2) (0.4, 0.4, 0.5) (0.5, 0.4, 0.1) (0.4, 0.3, 0.2) (0.4, 0.4, 0.1) (0.4, 0.2, 0.2)

S e1 e2 e2 e3 e1 e3

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soft relation S v2 v1 v2 v3 (0.4, 0.4, 0.5) (0.6, 0.3, 0.4) (0.4, 0.3, 0.2) (0.5, 0.3, 0.2) (0.4, 0.2, 0.2) (0.5, 0.3, 0.3)

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Table 6. Neutrosophic soft relation S.

Table 6: Neutrosophic soft relation S v5vv52v2 v5 vv65 v6 S S v3vv34v4 vv44vv55 e1 ee21 e2 (0.4, 0.2, 0.2) p0.4, (0.4,0.3, 0.3, 0.4) p0.3, (0.3, p0.4, 0.2,0.2) 0.2q (0.4, p0.4,0.4, 0.4, 0.2q 0.4q 0.2,0.2, 0.3q0.3) e2 ee32 e3 (0.6, 0.2, 0.1) p0.4, (0.4,0.3, 0.3, 0.2) p0.4, (0.4, p0.6, 0.2,0.4) 0.4q (0.3, p0.3,0.2, 0.2, 0.1q 0.2q 0.3,0.3, 0.4q0.4) p0.4, 0.2,0.3) 0.3q (0.4, p0.4,0.3, 0.3, 0.1q 0.2q 0.3,0.3, 0.5q0.5) 0.2, 0.1) p0.5, (0.5,0.3, 0.3, 0.2) p0.5, (0.5, e1 ee31 e3 (0.4, Let B “ Let B = SB “ SB =

tpe1 e2 , 0.4, 0.4, 0.5; q, pe2 e3 , 0.5, 0.4, 0.5q, pe1 e3 , 0.5, 0.2, 0.5qu be an NS on L, then {(e1 e2 , 0.4, 0.4, 0.5; ), (e2e3 , 0.5, 0.4, 0.5), (e1e3 , 0.5, 0.2, 0.5)} be a NS on L, then tpv v , 0.5, 0.4, 0.5q, pv v2 , 0.4, 0.2, 0.5q, pv2 v , 0.4, 0.2, 0.5q, pv2 v3 , 0.5, 0.3, 0.5q, {(v1 v11 ,10.5, 0.4, 0.5), (v1v1 2 , 0.4, 0.2, 0.5), (v2v11, 0.4, 0.2, 0.5), (v2v3 , 0.5, 0.3, 0.5), pv v , 0.5, 0.2, 0.5q, pv v , 0.4, 0.3, 0.5q, pv v , 0.5, 0.3, 0.5q, pv v , 0.5, 0.3, 0.5qu, (v3 v34 ,40.5, 0.2, 0.5), (v4v4 5 ,50.4, 0.3, 0.5), (v55v22, 0.5, 0.3, 0.5), (v55v66, 0.5, 0.3, 0.5)}, SB =“ {(v tpvv1 v,10.4, , 0.4,0.4, 0.4,0.5)(v 0.5qpvv1 v,20.5, , 0.5,0.4, 0.4,0.4), 0.4q,(v pv2vv1, ,0.5, 0.5,0.4, 0.4,0.4), 0.4q, (v pv2vv3,,0.4, 0.4, 0.4, 0.4, 0.5), 0.5q, SB 1 1 1 2 2 1 2 3 v , 0.4, 0.4, 0.5q, pv4 5v,50.4, , 0.4,0.4, 0.4,0.4), 0.4q,(v pv55vv22, ,0.4, 0.4,0.4, 0.4,0.5), 0.5q, (v pv55vv66,,0.4, 0.4, 0.4, 0.4, 0.5)}.

0.5qu. (vpv 3 v34 ,40.4, 0.4, 0.5), (v4 v

SBq is NSRR Hence, SpBq =“ (SB, pSB,SB) NSRRon onV́.

V́ .

Hence S(B)

Thus, G = (QA, SB) and G = (QA, SB) are LNAG and UNAG, respectively are shown in Figure 1.

Thus, G “ pQA, SBq and G “ pQA, SBq are LNAG and UNAG, respectively, are shown in Figure 1.

Hence, G = (G, G) is NSRG. (v1 , 0.5, 0.4, 0.5)

(v2 , 0.6, 0.3, 0.5) b

b

(0.5, 0.3, 0.5)

b

0.4, 0 .3,

(0 .5 ,0 .2 ,0 .5 ) ) 0.5 , .3 0 (0.5,

0.5) b

(v5 , 0.5, 0.4, 0.5)

b

(0.5, 0.4, 0.4)

) .5 ,0 .4 0 , .4 (0 b

b b

(v6 , 0.6, 0.3, 0.5)

(v3 , 0.7, 0.4, 0.5)

b

(0.4, 0 .4, 0.5)

(v3 , 0.5, 0.4, 0.6)

) 0.5 .4, 4, 0

(0 .4 ,0 .4 ,0 .5 )

b

4) 0.

4, . ,0 .4 (0

(v5 , 0.6, 0.4, 0.5)

(v4 , 0.6, 0.2, 0.5)

) .5 ,0 .3 ,0 .5 (0

(0.5, 0.4, 0.5)

(0.4, 0.2, 0.5)

(v2 , 0.5, 0.4, 0.6)

(v1 , 0.6, 0.4, 0.5) (0.4, 0.4, 0.5)

b

(v4 , 0.5, 0.4, 0.5)

(0. b (v6 , 0.6, 0.4, 0.5)

G = (QA, SB)

G = (QA, SB)

Figure Neutrosophic soft soft rough “= pG,(G, GqG) Figure 1:1.Neutrosophic roughgraph graphGG

Definition Let Gq G= (V, M, Q, S) be a neutrosophic soft rough graph on a non-empty set V . The Hence, 2.4.

G “ pG, is NSRG. order of G can be denoted by O(G), defined by Definition 4. Let G “ pV, M, O(G) Q, Sq be = a neutrosophic soft rough graph on a non-empty set V. The order of G O(G) + O(G), where X X can be denoted by OpGq, defined by O(G) = QA(v), O(G) = QA(v). v∈V

v∈V

OpGq “ OpGq ` OpGq, where ÿ denoted by S(G),ÿ The size of neutrosophic soft rough graph G, defined by OpGq “ QApvq, OpGq “ QApvq.

S(G) = (SG vPV+ SG), where v PV X X S(G) = SB(uv), S(G) = SB(uv).

The size of neutrosophic soft rough graph G, denoted by SpGq, defined by uv∈E

SpGq

“

SpGq

“

uv∈E

pSG ` SGq, where ÿ ÿ 6 SpGq “ SBpuvq, SBpuvq. uvPE

uvPE

Example 4. Let G be a neutrosophic soft rough graph as shown in Figure 1. Then, OpGq

“

p3.5, 2.0, 3.0q, OpGq “ p3.3, 2.4, 3.2q,

OpGq

“

OpGq ` OpGq “ p6.8, 4.4, 6.2q, and

SpGq

“

p3.2, 1.8, 3.0q SpGq “ p2.5, 2.4, 2.8q

SpGq

“

SpGq ` SpGq “ p5.7, 4.2, 5.8q.

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Definition 5. Let G1 “ pG1 , G1 q and G2 “ pG2 , G2 q be two neutrosophic soft rough graphs on V. The union of G1 and G2 is a neutrosophic soft rough graph G “ G1 Y G2 “ pG1 Y G2 , G1 Y G2 q, where G1 Y G2 “ pQA1 Y QA2 , SB1 Y SB2 q and G1 Y G2 “ pQA1 Y QA2 , SB1 Y SB2 q are neutrosophic graphs, such that (i)

@v P QA1 but v R QA2 .

TQA YQA2 pvq “TQA pvq, TQA1 YQA2 pvq “ TQA1 pvq, 1 1 IQA YQA2 pvq “IQA pvq, IQA1 YQA2 pvq “ IQA1 pvq, 1 1 FQA YQA2 pvq “FQA pvq, FQA1 YQA2 pvq “ FQA1 pvq.

1 1

(ii)

@v R QA1 but v P QA2 .

TQA YQA2 pvq “TQA2 pvq, TQA1 YQA2 pvq “ TQA2 pvq, 1 IQA YQA2 pvq “IQA2 pvq, IQA1 YQA2 pvq “ IQA2 pvq, 1 FQA YQA2 pvq “FQA2 pvq, FQA1 YQA2 pvq “ FQA2 pvq.

1

(iii) @v P QA1 X QA2 TQA YQA2 pvq “ maxtTQA pvq, TQA2 pvqu, TQA1 YQA2 pvq “ maxtTQA1 pvq, TQA2 pvqu, 1 1 IQA YQA2 pvq “ mintIQA pvq, IQA2 pvqu, IQA1 YQA2 pvq “ mintIQA1 pvq, IQA2 pvqu, 1 1 FQA YQA2 pvq “ mintFQA pvq, FQA2 pvqu, FQA1 YQA2 pvq “ mintFQA1 pvq, FQA2 pvqu.

1 1 (iv) @vu P SB1 but vu R SB2 .

TSB YSB2 pvuq “TSB pvuq, TSB1 YSB2 pvuq “ TSB1 pvuq, 1 1 ISB YSB2 pvuq “ISB pvuq, ISB1 YSB2 pvuq “ ISB1 pvuq, 1 1 FSB YSB2 pvuq “FSB pvuq, FSB1 YSB2 pvuq “ FSB1 pvuq.

1 1 (v)

@vu R SB1 but vu P SB2 TSB YSB2 pvuq “TSB2 pvuq, TSB1 YSB2 pvuq “ TSB2 pvuq, 1 ISB YSB2 pvuq “ISB2 pvuq, ISB1 YSB2 pvuq “ ISB2 pvuq, 1 FSB YSB2 pvuq “FSB2 pvuq, FSB1 YSB2 pvuq “ FSB2 pvuq.

1

(vi) @vu P SB1 X SB2 TSB YSB2 pvuq “ maxtTSB pvuq, TSB2 pvuqu, TSB1 YSB2 pvuq “ maxtTS B1 pvuq, TSB2 pvuqu, 1 1 ISB YSB2 pvuq “ mintISB pvuq, ISB2 pvuqu, ISB1 YSB2 pvuq “ mintISB1 pvuq, ISB2 pvuqu, 1 1 FSB YSB2 pvuq “ mintFSB pvuq, FSB2 pvuqu, FSB1 YSB2 pvuq “ mintFSB1 pvuq, FSB2 pvuqu.

1 1 Example 5. Let V “ tv1 , v2 , v3 , v4 u be a set of universes, and M “ te1 , e2 , e3 u a set of parameters. Then, a neutrosophic soft relation over V ˆ M can be written as in Table 7.

Table 7. Neutrosophic soft relation Q. Q

v1

v2

e1 e2 e3

p0.5, 0.4, 0.3q p0.3, 0.5, 0.6q p0.7, 0.5, 0.8q

p0.7, 0.6, 0.5q p0.4, 0.5, 0.1q p0.2, 0.3, 0.8q

v3 p0.7, 0.6, 0.4q p0.3, 0.6, 0.5q p0.7, 0.3, 0.5q

v4 p0.5, 0.7, 0.4q p0.4, 0.8, 0.2q p0.6, 0.4, 0.3q

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Let A1 “ tpe1 , 0.5, 0.7, 0.8q, pe2 , 0.7, 0.5, 0.3q, pe3 , 0.4, 0.5, 0.3qu, and A2 “ tpe1 , 0.6, 0.3, 0.5q, pe2 , 0.5, 0.8, 0.2q, pe3 , 0.5, 0.7, 0.2qu are two neutrosophic sets on M, Then, QpA1 q “ pQpA1 q, QpA1 qq and QpA2 q “ pQpA2 q, QpA2 qq are NSRSs, where QpA1 q

“

tpv1 , 0.5, 0.6, 0.5q, pv2 , 0.5, 0.5, 0.7qpv3 , 0.5, 0.5, 0.7q, pv4 0.4, 0.5, 0.5qu,

QpA1 q

“

tpv1 , 0.5, 0.5, 0.6q, pv2 , 0.5, 0.5, 0.3q, pv3 , 0.5, 0.5, 0.5q, pv4 0.5, 0.5, 0.3qu,

QpA2 q

“

tpv1 , 0.6, 0.5, 0.5q, pv2 , 0.5, 0.7, 0.5q, pv3 , 0.5, 0.7, 0.5q, pv4 , 0.5, 0.6, 0.5qu,

QpA2 q

“

tpv1 , 0.5, 0.4, 0.5q, pv2 , 0.6, 0.6, 0.2q, pv3 , 0.6, 0.6, 0.5q, pv4 , 0.5, 0.7, 0.2qu.

Let E “ tv1 v2 , v1 v4 , v2 v2 , v2 v3 , v3 v3 , v3 v4 u Ď V ˆ V, and L “ te1 e2 , e1 e3 , e2 e3 u Ă Ḿ. Then, a neutrosophic soft relation on E can be written as in Table 8.

Table 8. Neutrosophic soft relation S. S

v1 v2

v1 v4

v2 v2

v2 v3

v3 v3

v3 v4

e1 e2 e1 e3 e2 e3

(0.3, 0.4 ,0.1) (0.2 ,0.3 ,0.3) (0.2,0.3,0.5)

p0.4, 0.4, 0.2q p0.4, 0.3, 0.2q p0.3, 0.3, 0.3q

p0.4, 0.5, 0.1q p0.2, 0.3, 0.5q p0.2, 0.3, 0.1q

p0.3, 0.5, 0.4q p0.4, 0.3, 0.3q p0.4, 0.3, 0.1q

p0.3, 0.4, 0.4q p0.5, 0.3, 0.3q p0.3, 0.3, 0.5q

p0.4, 0.5, 0.2q p0.5, 0.4, 0.3q p0.3, 0.4, 0.3q

Let B1 “ tpe1 e2 , 0.5, 0.4, 0.5q, pe1 e3 , 0.3, 0.4, 0.5q, pe2 e3 , 0.4, 0.4, 0.3qu, and B2 “ tpe1 e2 , 0.5, 0.3, 0.2q, pe1 e3 , 0.4, 0.3, 0.3q, pe2 e3 , 0.4, 0.6, 0.2qu are two neutrosophic sets on L, Then, SpB1 q “ pSpB1 q, SpB1 qq and SpB2 q “ pSpB2 q, SpB2 qq are NSRRs, where SpB1 q

tpv1 v2 , 0.3, 0.4, 0.3q, pv1 v4 , 0.3, 0.4, 0.4q, pv2 v2 , 0.4, 0.4, 0.4q, pv2 v3 , 0.3, 0.4, 0.4q,

“

pv3 v3 , 0.3, 0.4, 0.5q, pv3 v4 , 0.3, 0.4, 0.5qu, SpB1 q

tpv1 v2 , 0.3, 0.4, 0.5q, pv1 v4 , 0.4, 0.4, 0.3q, pv2 v2 , 0.4, 0.4, 0.3q, pv2 v3 , 0.4, 0.4, 0.3q,

“

pv3 v3 , 0.3, 0.4, 0.5q, pv3 v4 , 0.4, 0.4, 0.3qu; SpB2 q

tpv1 v2 , 0.4, 0.6, 0.2q, pv1 v4 , 0.4, 0.6, 0.3q, pv2 v2 , 0.4, 0.6, 0.2q, pv2 v3 , 0.4, 0.6, 0.3q,

“

pv3 v3 , 0.4, 0.6, 0.3q, pv3 v4 , 0.4, 0.6, 0.3qu, SpB2 q

tpv1 v2 , 0.3, 0.3, 0.2q, pv1 v4 , 0.4, 0.3, 0.2q, pv2 v2 , 0.4, 0.3, 0.2q, pv2 v3 , 0.4, 0.3, 0.2q,

“

pv3 v3 , 0.4, 0.3, 0.3q, pv3 v4 , 0.4, 0.4, 0.2qu.

Thus, G1 “ pG1 , G1 q and G2 “ pG2 , G2 q are NSRGs, where G1 “ pQpA1 q, SpB1 qq, G1 “ pQpA1 q, SpB1 qq as shown in Figure 2.

(0.4, 0.4, 0.4)

(v1 , 0.5, 0.5, 0.6) b

(v1 , 0.5, 0.6, 0.5) b

(0.3, 0.4, 0.3) (v2 , 0.5, 0.5, 0.7)

(0 .3 ,0 .4 ,0 .4 )

(0.3, 0.4, 0.5) b

(v3 , 0.5, 0.5, 0.7)

, 0.4

) 0.3

(0.3, 0.4, 0.5) b

b

(0.4, 0.4, 0.3)

(v4 , 0.5, 0.5, 0.3)

(v2 , 0.5, 0.5, 0.3)

) .4 ,0 .4 0 3, 0.

(0.3, 0.4, 0.5) (

.4, (0

(0.3, 0.4, 0.5)

b

b

(v4 , 0.4, 0.5, 0.5)

G1

4, (0. b

(v3 , 0.5, 0.5, 0.5)

, 0.4

) 0.3

(0.4, 0.4, 0.3)

G1

Figure2:2. Neutrosophic Neutrosophic soft rough ,G Figure rough graph graph GG1 1“=pG (G 1 q1 ) 1 ,1G

G S(B2 )) as shown in Figure 3.

G22 = “ (Q(A pQpA22),q,S(B SpB22)), qq, G G22 = “ (Q(A pQpA2 ), 2 q, SpB2 qq as shown in Figure 3.

(0.4, 0.3, 0.2) (v2 , 0.6, 0.6, 0.2) b

(0.4, 0.6, 0.3)

(0 .3,

(v1 , 0.6, 0.5, 0.5) b

b

3) 0.

(0 .4 ,

0.6, 0.3)

(v4 , 0.5, 0.7, 0.2)

(0 .

0.3 ,

0.2 )

(0.4, 0.3, 0.2)

b

,0 (0.4

, 0.6, 0.2)

(v3 , 0.5, 0.7, 0.5)

b

(v1 , 0.5, 0.4, 0.5)

(v4 , 0.4, 0.5, 0.5) (v4 , 0.4, 0.5, 0.5)

0.3) .5, 0.3)

(v3 , 0.5, 0.5, 0.7) (v3 , 0.5, 0.5, 0.7)

(0.4, 0.4, 0.3) (0.4, 0.4, 0.3)

G G11

G1 G1

Figure 2: Neutrosophic soft rough graph G = (G , G ) Figure 2: Neutrosophic soft rough graph G11= (G11, G11) Axioms 2018, 7, 14

9 of 27

G = (Q(A ), S(B )), G = (Q(A ), S(B )) as shown in Figure 3.

G22= (Q(A22), S(B22)), G22 = (Q(A22), S(B22)) as shown in Figure 3.

(0.4, 0.6, 0.3) (0.4, 0.6, 0.3)

(v1 , 0.6, 0.5, 0.5) (v1 ,b 0.6, 0.5, 0.5)

(0.4, 0.3, 0.2) (0.4, 0.3, 0.2) (v2 , 0.6, 0.6, 0.2) b (v 2 , 0.6, 0.6, 0.2) b

(0 (0 .3, 0 .3, .3 0.3 , 0. , 0 2) .2)

b

b

b

(0 (0 .4, 0 .4 .6 ,0 , .6 0.3 ,0 ) .3 )

b

0.3) (0.4, 0.6, (0.4, 0.6, 0.3)

) .3 ,0) .60.3 , .06, 4 .0 (.04, (0

(v4 , 0.5, 0.7, 0.2) (v4 , 0.5,b 0.7, 0.2) b

(0.4, 0.3, 0.2) b (0.4, 0.3, 0.2) (v1 , 0.5,b 0.4, 0.5) (v1 , 0.5, 0.4, 0.5)

) , 0.2 ) , 0.3 , 0.2 (0.40.4, 0.3 (

0.2) (0.4, 0.6, (0.4, 0.6, 0.2)

(v3 , 0.5, 0.7, 0.5) (v3 , 0.5, 0.7, 0.5)

(0 (0 .4, .4 0. , 0 4, .4 0. , 0 2) .2 )

(0.4, 0.3, 0.3) (0.4, 0.3, 0.3) b

b

b (v2 , 0.5, 0.7, 0.5) b (v4 , 0.5, 0.6, 0.5) (v2 , 0.5, 0.7, 0.5) (v4 , 0.5, 0.6, 0.5) (0.4, 0.6, 0.2) (0.4, 0.6, 0.2) G G22

b

(v3 , 0.6, 0.6, 0.5) (v3 , 0.6, 0.6, 0.5)

G2 G2

Figure Neutrosophic soft soft rough pG(G q Figure 3:3.Neutrosophic roughgraph graphGG 2 “= 2, G 2 ,2 G2 ) Figure 3: Neutrosophic soft rough graph G22= (G 2 , G2 )

The union of GG11=“(G and = pG (G , G G22q) isisNSRG NSRGGG“=GG (G1Y∪GG2, ,GG1 1Y∪GG22q)asasshown shown union pG1,1,G ,GG1)1)qand andGG22 = “ Y∪GG2 2“=pG 11 TheThe union of of G1 = (G (G222,,G 1 2 2 ) is NSRG G = G1 ∪ G2 = (G11 ∪ G22 , G1 ∪ G2 ) as shown 1 ininFigure 4.

Figure4.4. in Figure

(0.4, 0.4, 0.2) (0.4, 0.4, 0.2)

(0.4, 0.4, 0.3) (0.4, 0.4, 0.3)

(0.4, 0.4, 0.3) (0.4, 0.4, 0.3)

(0.3, 0.3, 0.2) (0.3, 0.3, 0.2) (0 .(40 ,.04 .,30 ,.03 .,20 ).2 )

(v2 , 0.6, 0.5, 0.2) (v2 , 0.6, 0.5, 0.2)

b

b

b b

(v4 , 0.5, 0.5, 0.2) (v4 , 0.5, 0.5, 0.2)

b b

(v3 , 0.6, 0.5, 0.5) (v3 , 0.6, 0.5, 0.5)

0.2) (0.4, 0.3, (0.4, 0.3, 0.2)

G ∪G G11∪ G22

0.3) (0.4, 0.4, (0.4, 0.4, 0.3)

(0.4, 0.4, 0.3) b b (0.4, 0.4, 0.3) b (v4 ,b 0.5, 0.5, 0.5) (v3 , 0.5, 0.5, 0.5) (v4 , 0.5, 0.5, 0.5) (v3 , 0.5, 0.5, 0.5)

b

(0.4, 0.3, 0.3) (0.4, 0.3, 0.3)

) .2 ) , 0 .2 .3 , 0 , 0 .3 .4 , 0 (0 0.4 (

(v2 , 0.5, 0.5, 0.5) (v1 , 0.6, 0.5, 0.5) (v2 , 0.5, 0.5, 0.5) b 0.5, 0.5) (v1 , 0.6, b b (0.4, 0.4, 0.2) b (0.4, 0.4, 0.2) (0.4, 0.4, 0.2) (0.4, 0.4, 0.2)

(v1 , 0.5, 0.4, 0.5) (v1 , 0.5,b 0.4, 0.5)

G1 ∪ G2 G1 ∪ G2

∪G Figure 4: 4.

Neutrosophic soft rough graph GG1Y∪GG2“= (GY1 G 2, G Figure Neutrosophic softrough roughgraph graph Y 1G∪qG2 ) ,G Figure 4: Neutrosophic soft G11 ∪ G22 =pG (G 1 1 ∪ 2G 2 ,1G1 ∪2 G2 ) Definition 2.6. Let G1 = (G1 , G1 ) and G2 = (G2 , G2 ) be two NSRGs on V . The intersection of G1 Definition 2.6.LetLet = 1(G Gand (G two NSRGs onintersection V . The intersection , Gtwo Definition G1 G“1 pG , G11,qrough Ggraph be NSRGs on V. The of G1 and Gof a1 1 ) and 2 = 2 ) be 2 “GpG 2 q2G 2 isG 2 , G= and G2 is a6.neutrosophic soft G 1 ∩ G2 = (G1 ∩ G2 , G1 ∩ G2 ), where G1 ∩ G2 = G ∩ G ), where G and G is a neutrosophic soft rough graph G = G ∩ G = (G ∩ G , ∩ G = 1 2 2 1 2 1 2 1 X G 2 “ pQA1 X QA 12 , SB12 X neutrosophic soft rough graph G “ G1 X G2 “ pG1 X G2 , G1 X G2 q, where G SB2 q and G1 X G2 “ pQA1 X QA2 , SB1 X SB2 q are neutrosophic graphs, respectively, such that 9 9 (i) @v P QA1 but v R QA2 .

TQA XQA2 pvq “TQA pvq, TQA1 XQA2 pvq “ TQA1 pvq, 1 1 IQA XQA2 pvq “IQA pvq, IQA1 XQA2 pvq “ IQA1 pvq, 1 1 FQA XQA2 pvq “FQA pvq, FQA1 XQA2 pvq “ FQA1 pvq.

1 1 (ii)

@v R QA1 but v P QA2 .

TQA XQA2 pvq “TQA2 pvq, TQA1 XQA2 pvq “ TQA2 pvq, 1 IQA XQA2 pvq “IQA2 pvq, IQA1 XQA2 pvq “ IQA2 pvq, 1 FQA XQA2 pvq “FQA2 pvq, FQA1 XQA2 pvq “ FQA2 pvq.

1

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(iii) @v P QA1 X QA2 TQA XQA2 pvq “ mintTQA pvq, TQA2 pvqu, TQA1 XQA2 pvq “ mintTQA1 pvq, TQA2 pvqu, 1 1 IQA XQA2 pvq “ maxtIQA pvq, IQA2 pvqu, IQA1 XQA2 pvq “ maxtIQA1 pvq, IQA2 pvqu, 1 1 FQA XQA2 pvq “ maxtFQA pvq, FQA2 pvqu, FQA1 XQA2 pvq “ maxtFQA1 pvq, FQA2 pvqu.

1 1 (iv) @vu P SB1 but vu R SB2 .

TSB XSB2 pvuq “TSB pvuq, TSB1 XSB2 pvuq “ TSB1 pvuq, 1 1 ISB XSB2 pvuq “ISB pvuq, ISB1 XSB2 pvuq “ ISB1 pvuq, 1 1 FSB XSB2 pvuq “FSB pvuq, FSB1 XSB2 pvuq “ FSB1 pvuq.

1 1 (v)

@vu R SB1 but vu P SB2 TSB XSB2 pvuq “TSB2 pvuq, TSB1 XSB2 pvuq “ TSB2 pvuq, 1 ISB XSB2 pvuq “ISB2 pvuq, ISB1 XSB2 pvuq “ ISB2 pvuq, 1 FSB XSB2 pvuq “FSB2 pvuq, FSB1 XSB2 pvuq “ FSB2 pvuq.

1

(vi) @vu P SB1 X SB2 TSB XSB2 pvuq “ mintTSB pvuq, TSB2 pvuqu, TSB1 XSB2 pvuq “ mintTS B1 pvuq, TSB2 pvuqu, 1 1 ISB XSB2 pvuq “ maxtISB pvuq, ISB2 pvuqu, ISB1 XSB2 pvuq “ maxtISB1 pvuq, ISB2 pvuqu, 1 1 FSB XSB2 pvuq “ maxtFSB pvuq, FSB2 pvuqu, FSB1 XSB2 pvuq “ maxtFSB1 pvuq, FSB2 pvuqu.

1 1 Definition 7. Let G1 “ pG1 , G1 q and G2 “ pG2 , G2 q be two neutrosophic soft rough graphs on V. The join of G1 and G2 is a neutrosophic soft rough graph G “ G1 ` G2 “ pG1 ` G2 , G1 ` G2 q, where G1 ` G2 “ pQA1 ` QA2 , SB1 ` SB2 q and G1 ` G2 “ pQA1 ` QA2 , SB1 ` SB2 q are neutrosophic graph, respectively, such that (i)

@v P QA1 but v R QA2 .

TQA `QA2 pvq “TQA pvq, TQA1 `QA2 pvq “ TQA1 pvq, 1 1 IQA `QA2 pvq “IQA pvq, IQA1 `QA2 pvq “ IQA1 pvq, 1 1 FQA `QA2 pvq “FQA pvq, FQA1 `QA2 pvq “ FQA1 pvq.

1 1

(ii)

@v R QA1 but v P QA2 .

TQA `QA2 pvq “TQA2 pvq, TQA1 `QA2 pvq “ TQA2 pvq, 1 IQA `QA2 pvq “IQA2 pvq, IQA1 `QA2 pvq “ IQA2 pvq, 1 FQA `QA2 pvq “FQA2 pvq, FQA1 `QA2 pvq “ FQA2 pvq.

1

(iii) @v P QA1 X QA2 TQA `QA2 pvq “ maxtTQA pvq, TQA2 pvqu, TQA1 `QA2 pvq “ maxtTQA1 pvq, TQA2 pvqu, 1 1 IQA `QA2 pvq “ mintIQA pvq, IQA2 pvqu, IQA1 `QA2 pvq “ mintIQA1 pvq, IQA2 pvqu, 1 1 FQA `QA2 pvq “ mintFQA pvq, FQA2 pvqu, FQA1 `QA2 pvq “ mintFQA1 pvq, FQA2 pvqu.

1 1

Axioms 2018, 7, 14

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(iv) @vu P SB1 but vu R SB2 .

TSB `SB2 pvuq “TSB pvuq, TSB1 `SB2 pvuq “ TSB1 pvuq, 1 1 ISB `SB2 pvuq “ISB pvuq, ISB1 `SB2 pvuq “ ISB1 pvuq, 1 1 FSB `SB2 pvuq “FSB pvuq, FSB1 `SB2 pvuq “ FSB1 pvuq.

1 1 (v)

@vu R SB1 but vu P SB2 TSB `SB2 pvuq “TSB2 pvuq, TSB1 `SB2 pvuq “ TSB2 pvuq, 1 ISB `SB2 pvuq “ISB2 pvuq, ISB1 `SB2 pvuq “ ISB2 pvuq, 1 FSB `SB2 pvuq “FSB2 pvuq, FSB1 `SB2 pvuq “ FSB2 pvuq.

1

(vi) @vu P SB1 X SB2 TSB `SB2 pvuq “ maxtTSB pvuq, TSB2 pvuqu, TSB1 `SB2 pvuq “ maxtTS B1 pvuq, TSB2 pvuqu, 1 1 ISB `SB2 pvuq “ mintISB pvuq, ISB2 pvuqu, ISB1 `SB2 pvuq “ mintISB1 pvuq, ISB2 pvuqu, 1 1 FSB `SB2 pvuq “ mintFSB pvuq, FSB2 pvuqu, FSB1 `SB2 pvuq “ mintFSB1 pvuq, FSB2 pvuqu.

1 1

(vii) @vu P Ẽ, where Ẽ is the set of edges joining vertices of QA1 and QA2 .

TSB `SB2 pvuq “ mintTQA pvq, TQA2 puqu, TSB1 `SB2 pvuq “ mintTQA1 pvq, TQA2 puqu, 1 1 ISB `SB2 pvuq “ maxtIQA pvq, IQA2 puqu, ISB1 `SB2 pvuq “ maxtIQA1 pvq, IQA2 puqu, 1 1 FSB `SB2 pvuq “ maxtFQA pvq, FQA2 puqu, FSB1 `SB2 pvuq “ maxtFQA1 pvq, FQA2 puqu.

1 1 Definition 8. The Cartesian product of G1 and G2 is a G “ G1 ˙ G2 “ pG1 ˙ G2 , G1 ˙ G2 q, where G1 ˙ G2 “ pQA1 ˙ QA2 , SB1 ˙ SB2 q and G1 ˙ G2 “ pQA1 ˙ QA2 , SB1 ˙ SB2 q are neutrosophic digraph, such that (i)

@ pv1 , v2 q P QA1 ˆ QA2 .

TpQA ˙QA2 q pv1 , v2 q “ mintTQA pv1 q, TQA2 pv1 qu, TpQA1 ˙QA2 q pv1 , v2 q “ mintTQA1 pv1 q, TQA2 pv1 qu, 1 1 IpQA ˙QA2 q pv1 , v2 q “ maxtIQA pv1 q, IQA2 pv1 qu, IpQA1 ˙QA2 q pv1 , v2 q “ maxtIQA1 pv1 q, IQA2 pv1 qu, 1 1 FpQA ˙QA2 q pv1 , v2 q “ maxtFQA pv1 q, FQA2 pv1 qu, FpQA1 ˙QA2 q pv1 , v2 q “ maxtFQA1 pv1 q, FQA2 pv1 qu.

1 1

(ii)

@v1 v2 P SB2 , v P QA1 . ` ˘ TpSB ˙SB2 q pv, v1 qpv, v2 q “ mintTQA pvq, TSB2 pv1 v2 qu, 1 1 ` ˘ TpSB1 ˙SB2 q pv, v1 qpv, v2 q “ mintTQA1 pvq, TSB2 pv1 v2 qu, ` ˘ IpSB ˙SB2 q pv, v1 qpv, v2 q “ maxtIQA pvq, ISB2 pv1 v2 qu, 1 1 ` ˘ IpSB1 ˙SB2 q pv, v1 qpv, v2 q “ maxtIQA1 pvq, ISB2 pv1 v2 qu, ` ˘ FpSB ˙SB2 q pv, v1 qpv, v2 q “ maxtFQA pvq, FSB2 pv1 v2 qu, 1 1 ` ˘ FpSB1 ˙SB2 q pv, v1 qpv, v2 q “ maxtFQA1 pvq, FSB2 pv1 v2 qu.

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(iii) @v1 v2 P SB1 , v P QA2 . ` ˘ TpSB1 ˙SB2 q pv1 , vqpv2 , vq “ mintTSB1 pv1 v2 q, TQA2 pvqu, ` ˘ TpSB ˙SB2 q pv1 , vqpv2 , vq “ mintTSB pv1 v2 q, TQA2 pvqu, 1 1 ` ˘ IpSB ˙SB2 q pv1 , vqpv2 , vq “ maxtISB pv1 v2 q, IQA2 pvqu, 1 1 ` ˘ IpSB1 ˙SB2 q pv1 , vqpv2 , vq “ maxtISB1 pv1 v2 q, IQA2 pvqu, ` ˘ FpSB ˙SB2 q pv1 , vqpv2 , vq “ maxtFSB pv1 v2 q, FQA2 pvqu, 1 1 ` ˘ FpSB1 ˙SB2 q pv1 , vqpv2 , vq “ maxtFSB1 pv1 v2 q, FQA2 pvqu.

Definition 9. The cross product of G1 and G2 is a neutrosophic soft rough graph G “ G1 e G2 “ pG1 e G2 , G1 e G2 q, where G1 e G2 “ pQA1 e QA2 , SB1 e SB2 q and G1 e G2 “ pQA1 e QA2 , SB1 e SB2 q are neutrosophic graphs, respectively, such that (i)

@ pv1 , v2 q P QA1 ˆ QA2 .

TpQA eQA2 q pv1 , v2 q “ mintTQA pv1 q, TQA2 pv1 qu, TpQA1 eQA2 q pv1 , v2 q “ mintTQA1 pv1 q, TQA2 pv1 qu, 1 1 IpQA eQA2 q pv1 , v2 q “ maxtIQA pv1 q, IQA2 pv1 qu, IpQA1 eQA2 q pv1 , v2 q “ maxtIQA1 pv1 q, IQA2 pv1 qu, 1 1 FpQA eQA2 q pv1 , v2 q “ maxtFQA pv1 q, FQA2 pv1 qu, FpQA1 eQA2 q pv1 , v2 q “ maxtFQA1 pv1 q, FQA2 pv1 qu.

1 1

(ii)

@v1 u1 P SB1 , v2 u2 P SB2 . ` ˘ TpSB eSB2 q pv1 , v2 qpu1 , u2 q “ mintTSB pv1 u1 q, TSB2 pv1 u2 qu, 1 1 ` ˘ TpSB1 eSB2 q pv1 , v2 qpu1 , u2 q “ mintTSB1 pv1 u1 q, TSB2 pv1 u2 qu, ` ˘ IpSB eSB2 pv1 , v2 qpu1 , u2 q “ maxtISB pv1 u1 q, ISB2 pv1 u2 qu, 1 1 ` ˘ IpSB1 eSB2 q pv1 , v2 qpu1 , u2 q “ maxtISB1 pv1 u1 q, ISB2 pv1 u2 qu, ` ˘ FpSB eSB2 pv1 , v2 qpu1 , u2 q “ maxtFSB pv1 u1 q, FSB2 pv1 u2 qu, 1 1 ` ˘ FpSB1 eSB2 q pv1 , v2 qpu1 , u2 q “ maxtFSB1 pv1 u1 q, FSB2 pv1 u2 qu.

Definition 10. The rejection of G1 and G2 is a neutrosophic soft rough graph G “ G1 |G2 “ pG1 |G2 , G1 |G2 q, where G1 |G2 “ pSA1 |SA2 , SB1 |SB2 q and G1 |G2 “ pSA1 |SA2 , SB1 |SB2 q are neutrosophic graphs such that (i)

@ pv1 , v2 q P QA1 ˆ QA2 .

TpQA |QA2 q pv1 , v2 q “ mintTQA pv1 q, TQA2 pv2 qu, TpQA1 |QA2 q pv1 , v2 q “ mintTQA1 pv1 q, TQA2 pv2 qu, 1 1 IpQA |QA2 q pv1 , v2 q “ maxtIQA pv1 q, IQA2 pv2 qu, IpQA1 |QA2 q pv1 , v2 q “ maxtIQA1 pv1 q, IQA2 pv2 qu, 1 1 FpQA |QA2 q pv1 , v2 q “ maxtFQA pv1 q, FQA2 pv2 qu, FpQA1 |QA2 q pv1 , v2 q “ maxtFQA1 pv1 q, FQA2 pv2 qu.

1 1

(ii)

@v2 u2 R SB2 , v P QA1 . ` ˘ TpSB |SB2 q pv, v2 qpv, u2 q “ mintTQA pvq, TQA2 pv2 q, TQA2 pu2 qu, 1 1 ` ˘ TpSB1 |QB2 q pv, v2 qpv, u2 q “ mintTQA1 pvq, TQA2 pv2 q, TQA2 pu2 qu, ` ˘ pISB |SB2 q pv, v2 qpv, u2 q “ maxtIQA pvq, IQA2 pv2 q, IQA2 pu2 qu, 1 1 ` ˘ pISB1 |SB2 q pv, v2 qpv, u2 q “ maxtIQA1 pvq, IQA2 pv2 q, IQA2 pu2 qu, ` ˘ pFSB |SB2 q pv, v2 qpv, u2 q “ maxtFQA pvq, FQA2 pv2 q, FQA2 pu2 qu, 1 1 ` ˘ pFSB1 |SB2 q pv, v2 qpv, u2 q “ maxtFQA1 pvq, FQA2 pv2 q, FQA2 pu2 qu.

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(iii) @v1 u1 R SB1 , v P QA2 , ` ˘ TpSB1 |SB2 q pv1 , vqpu1 , vq “ mintTQA1 pv1 q, TQA1 pu1 q, TQA2 pvqu, ` ˘ IpSB1 |SB2 q pv1 , vqpu1 , vq “ maxtIQA1 pv1 q, IQA1 pu1 q, IQA2 pvqu, ` ˘ FpSB1 |SB2 q pv1 , vqpu1 , vq “ maxtFQA1 pv1 q, FQA1 pu1 q, FQA2 pvqu, ` ˘ TpSB |SB2 q pv1 , vqpu1 , vq “ mintTQA pv1 q, TQA pu1 q, TQA2 pvqu, 1 1 1 ` ˘ IpSB |SB2 q pv1 , vqpu1 , vq “ maxtIQA pv1 q, IQA pu1 q, IQA2 pvqu, 1 1 1 ` ˘ FpSB |SB2 q pv1 , vqpu1 , vq “ maxtFQA pv1 q, FQA pu1 q, FQA2 pvqu.

1

1

1

(iv) @v1 u1 R SB1 , v2 u2 R SB2 , v1 “ u1 . ` ˘ TpSB1 |SB2 q pv1 , v2 qpu1 , u2 q “ mintTQA1 pv1 q, TQA1 pu1 q, TQA2 pv2 q, TQA2 pu2 qu, ` ˘ IpSB1 |SB2 q pv1 , v2 qpu1 , u2 q “ maxtIQA1 pv1 q, IQA1 pu1 q, IQA2 pv2 q, IQA2 pu2 qu, ` ˘ FpSB1 |SB2 q pv1 , v2 qpu1 , u2 q “ maxtFQA1 pv1 q, FQA1 pu1 q, FQA2 pv2 q, FQA2 pu2 qu, ` ˘ TpSB |SB2 q pv1 , v2 qpu1 , u2 q “ mintTQA pv1 q, TQA pu1 q, TQA2 pv2 q, TQA2 pu2 qu, 1 1 1 ` ˘ IpSB |SB2 q pv1 , v2 qpu1 , u2 q “ maxtIQA pv1 q, IQA pu1 q, IQA2 pv2 q, IQA2 pu2 qu, 1 1 1 ` ˘ FpSB |SB2 q pv1 , v2 qpu1 , u2 q “ maxtFQA pv1 q, FQA pu1 q, FQA2 pv2 q, FQA2 pu2 qu, 1

1

1

Example 6. Let G1 “ pG1 , G1 q and G2 “ pG2 , G2 q be two neutrosophic soft rough graphs on V, where G1 “ pQA1 , SB1 q and G1 “ pQA1 , SB1 q are neutrosophic graphs as shown in Figure 2 and G2 “ pQA2 , SB2 q and G2 “ pQA2 , SB2 q are neutrosophic graphs as shown in Figure 3. The Cartesian product of G1 “ pG1 , G1 q and G2 “ pG2 , G2 q is NSRG G “ G1 ˆ G2 “ pG1 ˆ G2 , G1 ˆ G2 q as shown in Figure 5.

(0.4, 0.6, 0.5) (0.4, 0.6, 0.5) 0.5) 0.6, ((v1 , v2 ), 0.5, 0.7, 0.5) (0.4, ((v1 , v3 ), 0.5, 0.7, 0.5) b b

(0.4, 0.6, 0.5)

(0.4, 0.6, 0.7)

(0.3, 0.7, 0.5)

(0.3, 0.5, 0.5)

(0.4, 0.6, 0.5)

(0.4, 0.6, 0.7)

((v2 , v4 ), 0.5, 0.6, 0.7)

(0.4, 0.6, 0.5)

b

((v4 , v3 ), 0.4, 0.7, 0.5)

(0.4, 0.6, 0.5) (0.4, 0.6, 0.5)

(0.4, 0.6, 0.5)

(0.4, 0.6, 0.5) (0.4, 0.6, 0.5)

G1 × G2

(a) Figure 5. Cont.

(0 .4 ,0 .6 ,0 .5 )

(0.3, 0.6, 0.5)

b

(0.3, 0.6, 0.5)

(0.4, 0.6, 0.7)

(0 .3 ,0 .6 ,0 .5 )

(0.3, 0.7, 0.5)

(0.3, 0.7, 0.5)

((v4 , v2 ), 0.4, 0.7, 0.5)

((v3 , v4 ), 0.5, 0.6, 0.7)

((v3 , v3 ), 0.5, 0.7, 0.7) b

) , 0.5 , 0.7 (0.3

) , 0.5 , 0.7 (0.3

b

(0.3, 0.7, 0.5)

(0.3, 0.5, 0.5) (0.3, 0.5, 0.5)

b

(0.4, 0.6, 0.7)

b

(0.4, 0.6, 0.7)

(0.4, 0.6, 0.7) (0.3, 0.7, 0.5)

((v3 , v2 ), 0.5, 0.7, 0.7)

) .7 ,0 .6 ,0 .4 (0

((v4 , v1 ), 0.4, 0.5, 0.5)

((v2 , v3 ), 0.5, 0.7, 0.7)

(0.3, 0.6, 0.5)

b

(0.4, 0.6, 0.7)

(0.4, 0.6, 0.7)

b

) .7 ,0 .6 ,0 .4 (0

(0.3, 0.7, 0.5)

) .7 ,0 .6 ,0 .4 (0

b (0 .3 ,0 .5 ,0 .5 )

(0.4, 0.6, 0.7)

) .7 ,0 .6 ,0 .4 (0

((v3 , v1 ), 0.5, 0.5, 0.7)

((v2 , v2 ), 0.5, 0.7, 0.7) b

b

(0 .4 ,0 .5 ,0 .5 )

(0.3, 0.7, 0.5)

((v2 , v1 ), 0.5, 0.5, 0.7)

(0.3, 0.5, 0.5)

(0.4, 0.6, 0.7)

((v1 , v4 ), 0.5, 0.6, 0.5) b

(0.4, 0.6, 0.5)

(0.3, 0.6, 0.5)

b

(0.3, 0.7, 0.5)

((v1 , v1 ), 0.5, 0.6, 0.5)

((v4 , v4 ), 0.4, 0.6, 0.5) b

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(0.4, 0.5, 0.6) (0.4, 0.5, 0.6) 0.6) 0.5, ((v1 , v2 ), 0.5, 0.6, 0.6) (0.4, ((v1 , v3 ), 0.5, 0.6, 0.6) b b

(0.4, 0.5, 0.6)

b b

(0.4, 0.5, 0.3)

b

(0 .4 ,0 .7 ,0 .3 ) (0.4, 0.7, 0.3)

(0.4, 0.7, 0.3)

((v3 , v4 ), 0.5, 0.7, 0.5)

((v4 , v3 ), 0.5, 0.6, 0.5)

(0 .3 ,0 .7 ,0 .5 )

b

(0.4, 0.5, 0.5)

(0.4, 0.5, 0.3) (0.4, 0.5, 0.3)

(0.3, 0.7, 0.5)

(0.3, 0.6, 0.5)

((v4 , v2 ), 0.5, 0.6, 0.3)

(0.3, 0.5, 0.3)

(0.4, 0.5, 0.3)

(0.4, 0.7, 0.3)

) , 0.3 , 0.6 (0.4

(0.4, 0.4, 0.5)

((v4 , v1 ), 0.5, 0.5, 0.5)

(0.4, 0.5, 0.5)

b

((v3 , v3 ), 0.5, 0.6, 0.5) b

) .5 ,0 .5 ,0 .4 (0

(0.3, 0.5, 0.5)

((v2 , v4 ), 0.5, 0.7, 0.3)

((v2 , v3 ), 0.5, 0.6, 0.5)

(0.4, 0.5, 0.5) (0.3, 0.6, 0.5)

((v3 , v2 ), 0.5, 0.6, 0.5) b

) .5 ,0 .5 ,0 .4 (0

b (0 .3 ,0 .4 ,0 .5 )

(0.4, 0.6, 0.3)

(0.4, 0.4, 0.5)

(0.4, 0.4, 0.5)

(0.3, 0.6, 0.5)

((v3 , v1 ), 0.5, 0.5, 0.5)

(0.4, 0.5, 0.3)

b

) .3 ,0 .5 ,0 .4 (0

(0.3, 0.5, 0.3)

) .3 ,0 .5 ,0 .4 (0

(0 .4 ,0 .4 ,0 .5 )

(0.4, 0.6, 0.5)

((v2 , v2 ), 0.5, 0.6, 0.3) b

b

(0.4, 0.5, 0.3)

((v1 , v4 ), 0.5, 0.7, 0.6) b

(0.4, 0.5, 0.6)

(0.4, 0.6, 0.5)

(0.3, 0.6, 0.5)

(0.4, 0.6, 0.3)

((v2 , v1 ), 0.5, 0.5, 0.5)

(0.4, 0.6, 0.3)

(0.3, 0.4, 0.5)

(0.3, 0.5, 0.6)

(0.4, 0.6, 0.5)

b

) , 0.5 , 0.6 (0.4

((v1 , v1 ), 0.5, 0.5, 0.6)

((v4 , v4 ), 0.5, 0.7, 0.3) b

(0.4, 0.5, 0.3)

(0.4, 0.5, 0.3)

G1 × G2

(b) Figure 5: Cartesian product of two neutrosophic soft rough graphs G1 × G2 Figure 5. Cartesian product of two neutrosophic soft rough graphs G1 ˆ G2

Definition 2.11. The symmetric difference of G1 and G2 is a neutrosophic soft rough graph G = G1 ⊕ G2 = (G1 ⊕ G2 , G1 ⊕ G2 ), where G1 ⊕ G2 = (QA1 ⊕ QA2 , SB1 ⊕ SB2 ) and G1 ⊕ G2 = (QA1 ⊕ Definition 11. The symmetric difference of G1 and G2 is a neutrosophic soft rough graph G “ G1 ‘ G2 “ QA2 , SB1 ⊕ SB2 ) are neutrosophic graphs, respectively, such that pG1 ‘ G2 , G1 ‘ G2 q, where G1 ‘ G2 “ pQA1 ‘ QA2 , SB1 ‘ SB2 q and G1 ‘ G2 “ pQA1 ‘ QA2 , SB1 ‘ SB2 q (i)neutrosophic ∀ (v1 , v2 ) ∈graphs, QA1 ×respectively, QA2 . are such that (i)

T@(QA ) QA = min{T 1 , 1v2ˆ pv11,⊕QA v2 q 2P) (v QA QA1 (v1 ), TQA2 (v2 )}, T(QA1 ⊕QA2 ) (v1 , v2 ) = min{TQA1 (v1 ), TQA2 (v2 )}, 2.

I(QA1 ⊕QA2 ) (v1 , v2 ) = max{IQA1 (v1 ), IQA2 (v2 )}, I(QA1 ⊕QA2 ) (v1 , v2 ) = max{IQA1 (v1 ), IQA2 (v2 )}, TpQA ‘QA2 q pv1 , v2 q “ mintTQA pv1 q, TQA2 pv2 qu, TpQA1 ‘QA2 q pv1 , v2 q “ mintTQA1 pv1 q, TQA2 pv2 qu, 1 (v1 , v2 ) = max{FQA11 (v1 ), FQA2 (v2 )}, F(QA1 ⊕QA2 ) (v1 , v2 ) = max{FQA1 (v1 ), FQA2 (v2 )}.

F(QA 1 ⊕QA2 ) IpQA ‘QA2 q pv1 , v2 q “ maxtIQA pv1 q, IQA2 pv2 qu, IpQA1 ‘QA2 q pv1 , v2 q “ maxtIQA1 pv1 q, IQA2 pv2 qu, 1

1

(ii) ∀v v2 ∈ SB2pv , v1 , ∈ .

Fp1QA v2QA q “1maxtF QA1 pv1 q, FQA2 pv2 qu, FpQA1 ‘QA2 q pv1 , v2 q “ maxtFQA1 pv1 q, FQA2 pv2 qu.

1 ‘QA2 q

T(SB1 ⊕SB2 ) (v, v1 )(v, v2 ) = min{TQA1 (v), TSB2 (v1 v2 )}, (ii) @v1 v2 P SB2 , v P QA1 .

T(SB1 ⊕SB2 ) (v, v1 )(v, v2 ) = min{TQA1 (v), TSB2 (v1 v2 )}, ` ˘ pvq, T 2 pv v2 qu, IT(SB (v, vv11qpv, )(v, vv22q) “ = mintT max{IQA pSB11⊕SB ‘SB22q) pv, QA11 (v), ISB SB2 (v11 v2 )}, ` ˘ (v),TISB (v11vv22qu, )}, IT(SB (v, vv11qpv, )(v, vv22q) “ = mintT max{IQA SB22pv QA11pvq, pSB11⊕SB ‘SB22q) pv, ` ˘ F(SB (v),ISB FSB v2 )}, (v, vv11qpv, )(v, vv22q) “ = maxtI max{FQA pv, pvq, pv1(vv12 qu, IpSB1 ⊕SB QA 2 2 1 ‘SB22q) 11 ` ˘ (v, v )(v, v ) = max{FQApvq, F(SB FSB v2 )}.

IpSB11⊕SB pv2 1(vv12 qu, 1 (v),ISB ‘SB22q) pv, v11qpv, v22q “ maxtIQA 2 1 ` ˘ FpSB ‘SB2 q pv, v1 qpv, v2 q “ maxtFQA pvq, FSB2 pv1 v2 qu, 1 1 ` ˘ FpSB1 ‘SB2 q pv, v1 qpv, v2 q “ maxtFQA1 pvq, FSB2 pv1 v2 qu.

15

Axioms 2018, 7, 14

15 of 27

(iii) ∀v1 v2 ∈ SB1 , v ∈ QA2 .

(iii) @v1 v2 P SB1 , v P QA2 .

T(SB1 ⊕SB2 ) (v1 , v)(v2 , v) = min{TSB1 (v1 v2 ), TQA2 (v)}, ` ˘ pv1,,v)(v vqpv2,, vq “ pv1 v2 q, TQA pvqu,(v)}, TpSB ‘SB2 q (v T2QA T(SB v) =mintT min{TSBSB 1 1 1 (v1 v2 ), 2 2 1 ⊕SB2 ) ` 1 ˘ T pv v q, T pvqu, pv , vqpv , vq “ mintT 2 2 1 1 QA SB 2 1 ‘SB2 q (v1 , v)(v2 , v) = max{I 1 I(SBp1SB ⊕SB SB1 (v1 v2 ), IQA2 (v)}, 2) ` ˘ pv , vqpv , vq “ maxtI pv v q, I pvqu,(v)}, I 2 1 1 2 SBSB I(SBp1SB ), I2QA 2 q (v1 , v)(v2 , v) = max{I 1 ‘SB 1 1 (v1 v2 QA ⊕SB 2 2) ` ˘ I pv , vqpv , vq “ maxtI pv v q, I pvqu, 2, v) = max{F SB1 1(v2 v QA 2 1 ‘SB2 q (v11, v)(v2 F(SBp1SB ⊕SB SB1 1 2 ), FQA2 (v)}, 2) ` ˘ FpSB ‘SB2 q pv1 , vqpv2 , vq “ maxtFSB pv1 v2 q, FQA2 pvqu, 1 1 1 (v1 v2 ), FQA2 (v)}.

F(SB (v1 , v)(v2 , v) = max{FSB 1 ⊕SB2 ) ` ˘ FpSB1 ‘SB2 q pv1 , vqpv2 , vq “ maxtFSB1 pv1 v2 q, FQA2 pvqu.

(iv) ∀v1 u1 ∈ / SB1 , v2 u2 ∈ SB2 .

(iv) @v1 u1 R SB1 , v2 u2 P SB2 .

T(SB1 ⊕SB2 ) (v ` 1 , v2 )(u1 , u2 ) ˘ = min{TSB1 (v1 u1 ), TQA2 (v2 ), TQA2 (u2 )}, T SB ‘SB2 q pv1 , v2 qpu1 , u2 q “ mintTSB pv1 u1 q, TQA2 pv2 q, TQA2 pu2 qu, 1 1 T(SB1p⊕SB (v ` 1 , v2 )(u1 , u2 ) ˘ = min{TSB1 (v1 u1 ), TQA2 (v2 ), TQA2 (u2 )}, 2) TpSB1 ‘SB2 q pv1 , v2 qpu1 , u2 q “ mintTSB1 pv1 u1 q, TQA2 pv2 q, TQA2 pu2 qu, (v I(SB1 ⊕SB ` 1 , v2 )(u1 , u2 ) ˘ = max{ISB1 (v1 u1 ), IQA2 (v2 ), IQA2 (u2 )}, 2) IpSB ‘SB2 q pv1 , v2 qpu1 , u2 q “ maxtISB pv1 u1 q, IQA2 pv2 q, IQA2 pu2 qu, 1 1 I(SB1 ⊕SB (v ` 1 , v2 )(u1 , u2 ) ˘ = max{ISB1 (v1 u1 ), IQA2 (v2 ), IQA2 (u2 )}, 2) IpSB1 ‘SB2 q pv1 , v2 qpu1 , u2 q “ maxtISB1 pv1 u1 q, IQA2 pv2 q, IQA2 pu2 qu, F(SB1 ⊕SB2 ) (v ` 1 , v2 )(u1 , u2 ) ˘ = max{FSB1 (v1 u1 ), FQA2 (v2 ), FQA2 (u2 )}, FpSB ‘SB2 q pv1 , v2 qpu1 , u2 q “ maxtFSB pv1 u1 q, FQA2 pv2 q, FQA2 pu2 qu, 1 1 F(SB1 ⊕SB (v ` 1 , v2 )(u1 , u2 ) ˘ = max{FSB1 (v1 u1 ), FQA2 (v2 ), FQA2 (u2 )}.

2) FpSB1 ‘SB2 q pv1 , v2 qpu1 , u2 q “ maxtFSB1 pv1 u1 q, FQA2 pv2 q, FQA2 pu2 qu.

(v) ∀v1 u1 ∈ / SB1 , v2 u2 ∈ SB2 . (v)

@v1 u1 R SB1 , v2 u2 P SB2 .

` 1 , v2 )(u1 , u2 ) ˘ = min{T T(SB1 ⊕SB2 ) (v (v1 ), TQA1 (u1 ), TSB2 (v2 u2 )}, QA1pv TpSB ‘SB2 q pv1 , v2 qpu1 , u2 q “ mintTQA 1 q, TQA1 pu1 q, TSB2 pv2 u2 qu, 1 1 ` 1 , v2 )(u1 , u2 ) ˘ = min{TQA1 (v1 ), TQA1 (u1 ), TSB2 (v2 u2 )}, T(SB1 ⊕SB2 ) (v TpSB1 ‘SB2 q pv1 , v2 qpu1 , u2 q “ mintTQA1 pv1 q, TQA1 pu1 q, TSB2 pv2 u2 qu, ` 1 , v2 )(u1 , u2 ) ˘ = max{I I(SB1 ⊕SB2 ) (v 1 ), IQA (u1 ), ISB2 (v2 u2 )}, QA1 (vq, IpSB ‘SB2 q pv 2 qu, 1 , v2 qpu1 , u2 q “ maxtIQA1 pv 1 IQA1 pu11 q, ISB2 pv2 u 1 ` ˘ I(SB1 ⊕SB2 ) (v1 , v2 )(u1 , u2 ) = max{IQA1 (v1 ), IQA1 (u1 ), ISB2 (v2 u2 )}, IpSB1 ‘SB2 q pv1 , v2 qpu1 , u2 q “ maxtIQA1 pv1 q, IQA1 pu1 q, ISB2 pv2 u2 qu, ` 1 , v2 )(u1 , u2 ) ˘ = max{F F(SBF1 ⊕SB2 ) (v QA1 (v1 ), FQA1 (u1 ), FSB2 (v2 u2 )}, pSB1 ‘SB2 q pv1 , v2 qpu1 , u2 q “ maxtFQA1 pv1 q, FQA1 pu1 q, FSB2 pv2 u2 qu, ` 1 , v2 )(u1 , u2 ) ˘ = max{FQA (v1 ), FQA (u1 ), FSB (v2 u2 )}.

F(SBF1 ⊕SB2 ) (v 1 q, F pv , v qpu , u q “ maxtF pv pu1 q, F pv u2 qu.

1

pSB1 ‘SB2 q

2

1

2

QA1

1

1

QA1

2 2

SB2

) and = ,(G Example 2.7.7. Let G1 “= pG (G,1G , Gq1and 2 , G2 ) be two neutrosophic soft rough graphs on V , Example Let G G2 G “2 pG 1 1 1 2 G2 q be two neutrosophic soft rough graphs on V, where whereGG1“ pQA = (QA , SB ) and G = (QA , SB ) 1 1 1 1 1 are neutrosophic graphs as shown in Figure 6 and 1 , SB1 q and G1 “ pQA1 , SB1 q are neutrosophic graphs as shown in Figure 6 and G2 “ pQA2 , SB2 q 1 G = (QA2 , SB2 )graphs are neutrosophic graphs (QAG22, SB G2 = and 2 ) and “ pQA as shown in Figure 7. as shown in Figure 7 2 , SB2 q2are neutrosophic

.

(0

1) 0. b 3, 0. , .4 ,0 ) (a .2 ,0 .2 0 2,

b

) .3 ,0 ) .6 2 0 . , , 0 0.4 .2 , , 0 (a 4 .

(0 (0.1, 0.4, 0.2)

(0.3, 0.4, 0.3) b

b b

(b, 0.3, 0.2, , 0.3)

b

(b, 0.4, 0.5, 0.3)

(c, 0.5, 0.4, 0.3)

G1 = (QA1 , SB1 )

(c, 0.6, 0.2, 0.1)

G1 = (QA1 , SB1 )

Figure 6. Neutrosophicsoft soft rough G1 G “1pG G1 q , G1 ) Figure 6: Neutrosophic roughgraph graph =1 , (G 1

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) .3 b ,0 .3 0 ) , .3 b .4 ,0 ,0 .3 (b 0 ) , .3 .4 ,0 ,0 . 2 (b ) ,0 3 4 . . ,0 (0 .2 ,0 4 .

(0 (0.1, 0.3, 0.2) b

(a, 0.5, 0.3, 0.0)

b

) .3 b ,0 2) .6 0.

0 ) , , .3 .3 4 ) .

0 0 , , , 0 .2 .6 .4 (b , 0 0 (0 , .3 .4 ,0 ,0 .4 (b (0 (0.5, 0.3, 0.1) b b

b

(a, 1.0, 0.2, 0.9)

(c, 0.1, 0.2)0.3, 0.2) (0.1, 0.3,

b

(a, 0.5, 0.0) 2 , SB2 ) G20.3, = (QA

(0.5, 0.3, (c,0.1) 0.6, 0.3, 0.3)

b

b

b

(a, 1.0, G20.2, = 0.9) (QA2 , SB2 ) (c, 0.6, 0.3, 0.3)

(c, 0.1, 0.3, 0.2)

G2 = (QA2 , SB2 )

G2 = (QA2 , SB2 )

G22q) Figure 7: Neutrosophic soft rough Figure 7. Neutrosophic soft roughgraph graphGG22 = “ (G pG22,, G Figure 7: Neutrosophic soft rough graph G2 = (G2 , G2 ) The The symmetric difference of G G2 is G = G1 ⊕ G2 = (G1 ⊕ ‘ GG G ⊕G = G2G, G, G 1 ⊕‘ 2 ),q,where symmetric difference of1 Gand 2 where G11 ‘ G22 “ 1 and G2 is G “ G1 ‘ G2 “ pG 1 1 2 The difference of1 ⊕ G1Gand G2 is1 ⊕ G QA =G ⊕ 1G⊕ = (G G ⊕ G ), where G shown ⊕G , ⊕ G2 = QAsymmetric neutrosophic graphs as (QA1 ⊕ 2 SB 1 2 2 , SB1 ⊕ SB 2 ) and G 2 = (QA 2 ,1SB 2 ) 1are pQA1 ‘ QA2 , SB1 ‘ SB2 q and G1 ‘ G2 “ pQA1 ‘ QA2 , SB1 ‘ SB2 q are 2neutrosophic graphs as1 shown QA2 , SB1 ⊕ SB2 ) and G1 ⊕ G2 = (QA1 ⊕ QA2 , SB1 ⊕ SB2 ) are neutrosophic graphs as shown in (QA Figure 1 ⊕8. in Figure 8. in Figure 8.

(bb, 0.4, 0.5, 0.9)

4, (0.

0.3 )

0.3 )

4, 0.6 ,

(0.

4, 0.6 ,

(0.

(bb, 0.4, 0.5, 0.9)

) 0.3 6, , 0.)

4 .

(0 , 0.3 0.6

0.3 )

0.3 )

4, 0.6 ,

4, 0.6 ,

(0.

(0.

(bc, 0.4, 0.3, 0.3)

(bb, 0.4, 0.3, 0.3)

4, (0.

0.3 )

0.3 )

4, 0.4 ,

(0.

4, 0.4 ,

(0.

(bb, 0.4, 0.3, 0.3) (bc, 0.4, 0.3, 0.3)

.3) ,0 0.4 4, .3) .

0 ( ,0 0.4

0.3 )

0.3 )

1, 0.4 ,

(0.

1, 0.4 ,

(0.

(bc, 0.1, 0.3, 0.3)

b

(0.1, 0.6, 0.3)

(0.1, 0.6, 0.3)

(0.1, 0.4, 0.9)

(cc, 0.6, 0.3, 0.3)

(0.1, 0.4, 0.9)

(bc, 0.1, 0.3, 0.3)

b

b

(0.4, 0.6, 0.3)

(0.4, 0.6, 0.3)

(0.4, 0.2, 0.9)

G1 ⊕ G2

b

(cb, 0.4, 0.4, 0.3)

b

(0.1, 0.4, 0.3)

b

G ⊕ G2 0.4, 0.3) (cc, 0.1, 0.9, 0.3) 1 (ca, 0.5,

(ba, 0.4, 0.5, 0.9)

(0.4, 0.5, 0.3) (0.4, 0.5, 0.3) (ba, 0.4, 0.5, 0.9) b (0. (0.4, 0.5, 0.3) (0.4, 0.5, 0.3) 1, 0 .6, 0 (0. .3) 1, 0 .6, 0.3 ) .3) 6, 0 , 0.

1 .

(0 ) 3 .

0 .6, (0.4, 0.3, 0.2) (0.5,0.0.3, 1, 0 0.1) b b b ( (ca, 0.6, 0.2, 0.9) (cb, 0.3) (cc, 0.6, 0.3, 0.3) 0.6, 0.4, (0.4, 0.3, 0.2) (0.5, 0.3, 0.1) b b

(0.1, 0.4, 0.3)

(0.3, 0.4, 0.3)

b

b

(0.3, 0.4, 0.3)

(0.3, 0.4, 0.3)

(0.3, 0.4, 0.3)

b

(0.3, 0.3, 0.3) 1, 0(0.1, 0.3, 0.3) .4, 0 (0. .3) 1, 0 .4, 0.3 ) .3) 4, 0 , 0.

1 .

(0 ) 3 . ,0 0.4 0.3) (0.1, (0.4, 0.4, 0.3) 1,0.4, b b (0.

0.3) 0.4, 0.4, (cc, 0.1, 0.9, 0.3)(0.1,(ca, 0.4, 0.5, 0.3)0.4, 0.3) (cb, (0.4, 0.4, 0.3)

b

(0.4, 0.2, 0.9)

(0.3, 0.3, 0.3)

b

(0.4, 0.6, 0.3) 0.9) 0.4, 0.6, 0.3) (ab, 0.9) 0.6, 0.6,(0.4, ((ac, 0.4 0.1, 0.3, 0.2) (aa, 0.4, b .3) b b , 0.

6, 0 0.9) 6, 0(0.4, 0.6, 0.3) (0.4, 0.0.6, , .

4 3 .

(0.

(0 ) 3 4, 0 ) .

0 .6, .6, 0.3 4, 0 ) (0.

.3) ,0 0.)6 .4, .3 (0 , 0 0.6 .4, (0(0.4, 0.3, 0.3)

0.2, 0.3)

(ab, 0.4, 0.6, 0.3)

b

(0.4, 0.3, 0.3)

(0.2, 0.2, 0.3)

(0.2, 0.3, 0.2)

(0.

b

b

(0.2, 0.2, 0.3)

(0.2, 0.3, 0.2)

b

(0.1, 0.4, 0.2)

b

(ba, 0.3, 0.2, 0.3)

(0.1, 0.3, 0.3) (ba, 0.3,

(0.1, 0.4, 0.2)

9

.3) ,0 0.)4 .1, .3 (0 , 0 0.4 .1, (0(0.1, 0.3, 0.2)

b

b b

b

(0.1, 0.3, 0.2) (0.4, 0.3, 0.3) 0.4, 0.3, 0.3) (ab, 0.3, 0.1) ((ac, 0.1 0.1, 0.3, 0.2) (aa, 0.4, .2) b b b , 0.

3, 0 0.3) 3, 0(0.1, 0.3, 0.2) .0.3, (0.4, 0 , .

1 2 (0.

(0. ) 1, 0 ) 0.2 , .3, 3 .

0.2 1, 0 ) (0.

(0.1, 0.3, 0.2)

9

(ac, 0.1, 0.3, 0.2) (aa, 0.4, 0.6, 0.9)

(ab, 0.4, 0.3, 0.3)

(ac, 0.1, 0.3, 0.2) (aa, 0.4, 0.3, 0.1) b

b

G1(ca, ⊕G 2 0.2, 0.9) (cb, 0.4, 0.6, 0.3) 0.6, G1 ⊕ G2

Figure 8: Neutrosophic soft rough graph G1 ⊕ G2 = (G1 ⊕ G2 , G1 ⊕ G2 ) ,G ‘G q Figure Neutrosophicsoft softrough rough graph pG1 ‘ G 2 “ Figure 8: 8.

Neutrosophic graph GG11‘⊕GG 2 = (G 1 ⊕2 G21, G1 2⊕ G2 ) Definition 2.12. The lexicographic product of G1 and G2 is a neutrosophic soft rough graph G = ∗ ∗ 2.12. product ofGG G2 1is⊙asoft neutrosophic G1Definition ⊙ G2 = (G12. ), where = (QA QA ⊙ SB ) and ⊙ pG G∗21 ˚G =d = TheGThe lexicographic of GG neutrosophic rough Gsoft Grough G∗12 graph “ Definition 1∗ ⊙ 2 ∗ , Glexicographic 21∗aand 2 , SB1graph 2“ ∗ ⊙ 1 ⊙ G2product 2Gis 1 1and 1 dG ∗ ∗ ∗ ∗ ˚ ˚ ˚ ˚ G ⊙ G = (G ⊙ G , G ⊙ G ), where G ⊙ G = (QA ⊙ QA , SB ⊙ SB ) and G ⊙ G 2 2,G 1⊙ 1 1 d QA 2 2 , SB1 d1SB2 q are QA SB are neutrosophic respectively, (QAG112 ˚ , G ∗ SBG 2 = d 1G QA12 ∗, SB G11 dthat G2 2“ pQA 2 )21∗˚ 2 ˚ “2pQA1 dgraphs, 1 d2 ∗SB2 q andsuch 2 q,1 where 1 d SB1 ⊙ respectively, SB2 ) are neutrosophic (QA 1 ⊙ QA2 ,graphs, neutrosophic such that graphs, respectively, such that (i) ∀ (v1 , v2 ) ∈ QA1 × QA2 . ∀@ (v ∈ QA1 ˆ × QA QA2 .. (i)(i) pv11,, vv22)q(v P QA 2 1 T(QA 1 , v2 ) = min{TQA1 (v1 ), TQA2 (v2 )}, T(QA1 ⊙QA2 ) (v1 , v2 ) = min{TQA1 (v1 ), TQA2 (v2 )}, 1 ⊙QA2 ) T(QA1 ⊙QA2 ) (v1 , v2 ) = min{TQA1 (v1 ), TQA2 (v2 )}, T(QA1 ⊙QA2 ) (v1 , v2 ) = min{TQA1 (v1 ), TQA2 (v2 )}, )},qu, I(QA ), Iq,QA (v2 )}, I T 1 ⊙QA2 ) (v1pv (v1 , v2 ) = max{IQA1 (v1pv , v2 ) = max{IQA1 (v1pv), Iq,QA 2 (v2 TpQA pv 1 ⊙QA2 ) 2 1 T2QA pv2 qu,(QA 1 TQA pQA dQA q 1 , v2 q “ mintTQA 2 dQA q pv1 , v2 q “ mintT QA 1 I(QA11⊙QA22) (v1 , v2 ) = max{IQA11 (v1 ), IQA22 (v2 )}, I(QA11 ⊙QA22 ) (v1 , v2 ) = max{IQA1 (v1 ), IQA2 (v2 )}, F(QA1 ⊙QA2 ) (v1 , v2 ) = max{FQA1 (v1 ), FQA2 (v2 )}, F(QA1 ⊙QA2 ) (v1 , v2 ) = max{FQA1 (v1 ), FQA2 (v2 )}.

IpQA dQA2 q pv1 , v2 q “ maxtIQA pv1 q, IQA2 pv2 qu, IpQA1 dQA2 q pv1 , v2 q “ maxtIQA1 pv1 q, IQA2 pv2 qu, 1 F(QA (v1 , v2 ) = max{FQA11 (v1 ), FQA2 (v2 )}, F(QA1 ⊙QA2 ) (v1 , v2 ) = max{FQA1 (v1 ), FQA2 (v2 )}.

1 ⊙QA2 ) pv∈1 , QA v2 q 1“ (ii) ∀v1Fvp2QA ∈1SB . maxtFQA1 pv1 q, FQA2 pv2 qu, FpQA1 dQA2 q pv1 , v2 q “ maxtFQA1 pv1 q, FQA2 pv2 qu.

2 ,2 qv dQA (ii) ∀v1 v2 ∈ SB2 , v ∈ QA1 .

(v, v1 )(v, v2 ) = min{TQA1 (v), TSB2 (v1 v2 )}, T(SB ⊙SB ) (ii) @v1 v2 P SB2 , v P QA .

1 2 1 T 1 ⊙SB2 ) (v, v1 )(v, v 2 ) = min{TQA1 (v), TSB2 (v1 v2 )}, (v, T(SB(SB 1 ⊙SB2 ) ` v1 )(v, v2 ) = ˘ min{TQA1 (v), TSB2 (v1 v2 )},

pv, v qpv, v q pv(v (v, v )(v, v ) =mintT min{TQA (v),TTSB TT(SB 11 v ) 22 = “ 1 v2vqu, QA11pvq, 2 )}, ⊙SB SB22q) v1 )(v, 2 21 v21)}, 11d I(SBp1SB ISB2SB (v 2 ⊙SB2 ) (v, max{IQA1 (v), ` ˘ IT(SB (v,vv11qpv, )(v, v ) =mintT max{IQA11pvq, (v),TISB )}, pv(v1vv2qu, ⊙SB SB SB22(v, q) pv, pSB 22 1 2 v1 )(v, v2v) 22q= “ max{IQAQA I(SB 11d 1 (v), ISB2 (v1 v2 )}, 1 ⊙SB2 ) ` ˘

(v, v11qpv, )(v, v ) =maxtI max{IQA1pvq, (v), ISBpv I 1 ⊙SB2 ) pv, 2 )}, 2 (v v122v)}, qu, 1v F(SBI(SB (v), FISB (v, )(v, v2v) 22q= “ max{FQA p1SB q v1v QA SB22 (v 1 1 dSB ⊙SB 2) 2 1 1 ` ˘ F 1 ⊙SB2 ) pv, (v), F pv2 (v v2 )}, (v, v1qpv, )(v, v 2q) “ =maxtI max{FQApvq, 1qu, 1 FISBSB (v, v1v)(v, ) 2 = max{F F(SBI(SB 1 v2v QA QA SB2 (v1 v2 )}. p1SB ⊙SB 1 (v), 1 2) 2q 1 dSB F(SB1 ⊙SB2 )` (v, v1 )(v, v2 )˘ = max{FQA1 (v), FSB2 (v1 v2 )}.

FpSB dSB2 q pv, v1 qpv, v2 q “ maxtFQA pvq, FSB2 pv1 v2 qu, 1 1 ` ˘ 17 FpSB1 dSB2 q pv, v1 qpv, v2 q “ maxtFQA1 pvq, FSB2 pv1 v2 qu.

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(iii) @v1 u1 P SB1 , v1 u2 P SB2 . ` ˘ TpSB dSB2 q pv1 , v1 qpu1 , u2 q “ mintTSB pv1 u1 q, TSB2 pv1 u2 qu, 1 1 ` ˘ TpSB1 dSB2 q pv1 , v1 qpu1 , u2 q “ mintTSB1 pv1 u1 q, TSB2 pv1 u2 qu, ` ˘ IpSB dSB2 pv1 , v1 qpu1 , u2 q “ maxtISB pv1 u1 q, ISB2 pv1 u2 qu, 1 1 ` ˘ IpSB1 dSB2 q pv1 , v1 qpu1 , u2 q “ maxtISB1 pv1 u1 q, ISB2 pv1 u2 qu, ` ˘ FpSB dSB2 pv1 , v1 qpu1 , u2 q “ maxtFSB pv1 u1 q, FSB2 pv1 u2 qu, 1 1 ` ˘ FpSB1 dSB2 q pv1 , v1 qpu1 , u2 q “ maxtFSB1 pv1 u1 q, FSB2 pv1 u2 qu.

Definition 13. The strong product of G1 and G2 is a neutrosophic soft rough graph G “ G1 b G2 “ pG1 ˚ b G2 ˚ , G1˚ b G2˚ q, where G1 ˚ b G2 ˚ “ pQA1 b QA2 , SB1 b SB2 q and G1˚ b G2˚ “ pQA1 b QA2 , SB1 b SB2 q are neutrosophic graphs, respectively, such that (i)

@ pv1 , v2 q P QA1 ˆ QA2 .

TpQA bQA2 q pv1 , v2 q “ mintTQA pv1 q, TQA2 pv2 qu, TpQA1 bQA2 q pv1 , v2 q “ mintTQA1 pv1 q, TQA2 pv2 qu, 1 1 IpQA bQA2 q pv1 , v2 q “ maxtIQA pv1 q, IQA2 pv2 qu, IpQA1 bQA2 q pv1 , v2 q “ maxtIQA1 pv1 q, IQA2 pv2 qu, 1 1 FpQA bQA2 q pv1 , v2 q “ maxtFQA pv1 q, FQA2 pv2 qu, FpQA1 bQA2 q pv1 , v2 q “ maxtFQA1 pv1 q, FQA2 pv2 qu.

1 1

(ii)

@v1 v2 P SB2 , v P QA1 . ` ˘ TpSB bSB2 q pv, v1 qpv, v2 q “ mintTQA pvq, TSB2 pv1 v2 qu, 1 1 ` ˘ TpSB1 bSB2 q pv, v1 qpv, v2 q “ mintTQA1 pvq, TSB2 pv1 v2 qu, ` ˘ IpSB bSB2 q pv, v1 qpv, v2 q “ maxtIQA pvq, ISB2 pv1 v2 qu, 1 1 ` ˘ IpSB1 bSB2 q pv, v1 qpv, v2 q “ maxtIQA1 pvq, ISB2 pv1 v2 qu, ` ˘ FpSB bSB2 q pv, v1 qpv, v2 q “ maxtFQA pvq, FSB2 pv1 v2 qu, 1 1 ` ˘ FpSB1 bSB2 q pv, v1 qpv, v2 q “ maxtFQA1 pvq, FSB2 pv1 v2 qu.

(iii) @v1 v2 P SB1 , v P QA2 . ` ˘ TpSB bSB2 q pv1 , vqpv2 , vq “ mintTSB pv1 v2 q, TQA2 pvqu, 1 1 ` ˘ TpSB1 bSB2 q pv1 , vqpv2 , vq “ mintTSB1 pv1 v2 q, TQA2 pvqu, ` ˘ IpSB bSB2 q pv1 , vqpv2 , vq “ maxtISB pv1 v2 q, IQA2 pvqu, 1 1 ` ˘ IpSB1 bSB2 q pv1 , vqpv2 , vq “ maxtISB1 pv1 v2 q, IQA2 pvqu, ` ˘ FpSB bSB2 q pv1 , vqpv2 , vq “ maxtFSB pv1 v2 q, FQA2 pvqu, 1 1 ` ˘ FpSB1 bSB2 q pv1 , vqpv2 , vq “ maxtFSB1 pv1 v2 q, FQA2 pvqu.

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(iv) @v1 u1 P SB1 , v1 u2 P SB2 . ` ˘ TpSB bSB2 q pv1 , v1 qpu1 , u2 q “ mintTSB pv1 u1 q, TSB2 pv1 u2 qu, 1 1 ` ˘ TpSB1 bSB2 q pv1 , v1 qpu1 , u2 q “ mintTSB1 pv1 u1 q, TSB2 pv1 u2 qu, ` ˘ IpSB bSB2 pv1 , v1 qpu1 , u2 q “ maxtISB pv1 u1 q, ISB2 pv1 u2 qu, 1 1 ` ˘ IpSB1 bSB2 q pv1 , v1 qpu1 , u2 q “ maxtISB1 pv1 u1 q, ISB2 pv1 u2 qu, ` ˘ FpSB bSB2 pv1 , v1 qpu1 , u2 q “ maxtFSB pv1 u1 q, FSB2 pv1 u2 qu, 1 1 ` ˘ FpSB1 bSB2 q pv1 , v1 qpu1 , u2 q “ maxtFSB1 pv1 u1 q, FSB2 pv1 u2 qu.

Definition 14. The composition of G1 and G2 is a neutrosophic soft rough graph G “ G1 rG2 s “ pG1 ˚ rG2 ˚ s, G1˚ rG2˚ sq, where G1 ˚ rG2 ˚ s “ pQA1 rQA2 s, SB1 rSB2 sqs and G1˚ rG2˚ s “ pQA1 rQA2 s, SB1 rSB2 sq are neutrosophic graphs, respectively, such that (i)

@pv1 , v2 q P QA1 ˆ QA2 .

TpQA ˆQA2 q pv1 , v2 q “ mintTQA pv1 q, TQA2 pv2 qu, TpQA1 ˆQA2 q pv1 , v2 q “ mintTQA1 pv1 q, TQA2 pv2 qu, 1 1 IpQA ˆQA2 q pv1 , v2 q “ maxtIQA pv1 q, IQA2 pv2 qu, IpQA1 ˆQA2 q pv1 , v2 q “ maxtIQA1 pv1 q, IQA2 pv2 qu, 1 1 FpQA ˆQA2 q pv1 , v2 q “ maxtFQA pv1 q, FQA2 pv2 qu, FpQA1 ˆQA2 q pv1 , v2 q “ maxtFQA1 pv1 q, FQA2 pv2 qu.

1 1

(ii)

@v1 v2 P SB2 , v P QA1 . ` ˘ TpSB ˆSB2 q pv, v1 qpv, v2 q “ mintTQA pvq, TSB2 pv1 v2 qu, 1 1 ` ˘ TpSB1 ˆSB2 q pv, v1 qpv, v2 q “ mintTQA1 pvq, TSB2 pv1 v2 qu, ` ˘ IpSB ˆSB2 q pv, v1 qpv, v2 q “ maxtIQA pvq, ISB2 pv1 v2 qu, 1 1 ` ˘ IpSB1 ˆSB2 q pv, v1 qpv, v2 q “ maxtIQA1 pvq, ISB2 pv1 v2 qu, ` ˘ FpSB ˆSB2 q pv, v1 qpv, v2 q “ maxtFQA pvq, FSB2 pv1 v2 qu, 1 1 ` ˘ FpSB1 ˆSB2 q pv, v1 qpv, v2 q “ maxtFQA1 pvq, FSB2 pv1 v2 qu.

(iii) @v1 v2 P SB1 , v P QA2 . ` ˘ TpSB ˆSB2 q pv1 , vqpv2 , vq “ mintTSB pv1 v2 q, TQA2 pvqu, 1 1 ` ˘ TpSB1 ˆSB2 q pv1 , vqpv2 , vq “ mintTSB1 pv1 v2 q, TQA2 pvqu, ` ˘ IpSB ˆSB2 q pv1 , vqpv2 , vq “ maxtISB pv1 v2 q, IQA2 pvqu, 1 1 ` ˘ IpSB1 ˆSB2 q pv1 , vqpv2 , vq “ maxtISB1 pv1 v2 q, IQA2 pvqu, ` ˘ FpSB ˆSB2 q pv1 , vqpv2 , vq “ maxtFSB pv1 v2 q, FQA2 pvqu, 1 1 ` ˘ FpSB1 ˆSB2 q pv1 , vqpv2 , vq “ maxtFSB1 pv1 v2 q, FQA2 pvqu.

Axioms 2018, 7, 14

19 of 27

(iv) @v1 u1 P SB1 , v1 ‰ u2 P QA2 . ` ˘ TpSB ˆSB2 q pv1 , v1 qpu1 , u2 q “ mintTSB pv1 u1 q, TSB2 pv1 u2 qu, 1 1 ` ˘ TpSB1 ˆSB2 q pv1 , v1 qpu1 , u2 q “ mintTSB1 pv1 u1 q, TSB2 pv1 u2 qu, ` ˘ IpSB ˆSB2 pv1 , v1 qpu1 , u2 q “ maxtISB pv1 u1 q, ISB2 pv1 u2 qu, 1 1 ` ˘ IpSB1 ˆSB2 q pv1 , v1 qpu1 , u2 q “ maxtISB1 pv1 u1 q, ISB2 pv1 u2 qu, ` ˘ FpSB ˆSB2 pv1 , v1 qpu1 , u2 q “ maxtFSB pv1 u1 q, FSB2 pv1 u2 qu, 1 1 ` ˘ FpSB1 ˆSB2 q pv1 , v1 qpu1 , u2 q “ maxtFSB1 pv1 u1 q, FSB2 pv1 u2 qu.

Definition 15. Let G “ pG, Gq be a neutrosophic soft rough graph. The complement of G, denoted by ´ SBq ´ is a neutrosophic soft rough graph, where Ǵ “ pQA, ´ “ pQA, ´ are neutrosophic ´ SBq ´ and G Ǵ “ pǴ, Gq graphs such that (i)

@v P QA. ´ pvq “ T ´ ´ TQA QApvq , IQA pvq “ IQApvq , FQA pvq “ FQApvq , ´ pvq “ TQApvq , IQA ´ pvq “ IQApvq , FQA ´ pvq “ FQApvq .

TQA

(ii) ∀ v, u ∈ QA. (ii)

@ v, u P QA.

´ (vu) = min{T (v), T (u)} − T (vu), TSB QA QA SB

´ pvuq “ mintT pvq, T puqu ´ T pvuq, TSB ´ QA QA SB

ISB (vu) = max{IQA (v), IQA (u)} − ISB (vu),

´ pvuq “ maxtI pvq, I puqu ´ I pvuq, ISB QA QA SB

F ´ (vu) = max{F

(u)} − FSB (vu),

(v), F

QA QA ´SB pvuq “ maxtFQA pvq, FQA puqu ´ FSB pvuq, FSB

´

T´SB (vu) = min{TQA (v), TQA (u)} − TSB (vu), TSB pvuq “ mintTQA pvq, TQA puqu ´ TSB pvuq, ´

(vu) = max{I (v), IQA − I (vu), ´SB QA IISB pvuq “ maxtI IQA puqu ´ I(u)} QA pvq, SB pvuq, SB

´pvuq ´SB (vu) = max{F (v), FQA − FSB (vu).

FFSB “ maxtF FQA puqu ´ F(u)} QA QA pvq, SB pvuq. ´ is Example 8. Consider an NSRGs G as shown Figure 9.in Figure 9. The complement of G is Ǵ = (Ǵ, G) Example 2.8.

Consider a NSRGs G asinshown (u, 0.4, 0.1, 0.2)

b

(0.1, 0.0, 0.1)

.3, 2, 0 (0.

) 0.1

b

(u, 0.5, 0.3, 0.1)

b

b

(0.1, 0.3, 0.5)

.3, 1, 0 (0. b

(w, 0.3, 0.1, 0.1)

(x, 0.2, 0.3, 0.3)

(v, 0.2, 0.3, 0.5)

b

(0.2, 0.3, 0.1)

(0.1, 0.0, 0.1)

(v, 0.5, 0.3, 0.1)

) 0.0

b b

(x, 0.3, 0.3, 0.2)

(w, 0.3, 0.0, 0.1)

G = (QA, SB)

G = (QA, SB)

Figure 9. Neutrosophic soft rough graph G “ pG, Gq Figure 9: Neutrosophic soft rough graph G = (G, G)

´

´ SBq ´ and The complement of G is Ǵ “ pǴ, Gq is obtained by using where ´15, ´ ǴSB) ´“ pQA, ´ the ´ Definition obtained by´ using the Definition 2.15, where Ǵ = (QA, SB) and G = (QA, are neutrosophic graphs ´ ´ G “ pin QA, SBq are10. neutrosophic graphs as shown in Figure 10. as shown Figure (0.2, 0.0, 0.1)

3, 0 .

5) (0.3, 0.3, 0.2)

(x, 0.3, 0.3, 0.2)

(0.3, 0.3, 0.1)

2, 0 .

(0.1, 0.0, 0.1)

(0.3 , 0.3 , 0.2 )

(0.3, 0.1, 0.2)

(0.2 , 0.3 , 0.3 )

(0.1, 0.0, 0.1)

G = (QA, SB)

b

(0.

) 0.1

b (0.3, 0.1, 0.1) (w, 0.3, 0.1, 0.1)

(u, 0.5, 0.3,b 0.1) (0.2, 0.3, 0.5)

, 0.3

(x, 0.2, 0.3, 0.3)

b

.5, (0

1) (0.2, 0.3, 0.3)

(0.2, 0.3, 0.5)

3, 0 .

) 0.2

3, 0 .

(0.2, 0.3, 0.5)

(0.4, 0.3, 0.2)

(v, 0.2, 0.3, 0.5)

, 0.1

(0.2, 0.3, 0.3)

(0.5, 0.3, 0.1)

b

(u, 0.4, 0.1,b 0.2)

b

(0.

(0.1, 0.0, 0.0)

.4, (0

(v, 0.5, 0.3, 0.1)

b (0.3, 0.0, 0.1) (w, 0.3, 0.0, 0.1)

G = (QA, SB)

´ Figure 10: Neutrosophic soft rough graph Ǵ = (Ǵ, G)

G = (QA, SB)

G = (QA, SB)

Figure 9: Neutrosophic soft rough graph G = (G, G) ´ = (QA, ´ SB) ´ are neutrosophic ´ SB) ´ and G Axioms 2018,the 7, 14 Definition 2.15, where Ǵ = (QA, 20 of 27 obtained by using graphs as shown in Figure 10.

(0.2, 0.0, 0.1)

3, 0 .

5) (0.3, 0.3, 0.2)

(x, 0.3, 0.3, 0.2)

G = (QA, SB)

(0.3, 0.3, 0.1)

2, 0 .

(0.1, 0.0, 0.1)

(0.3 , 0.3 , 0.2 )

(0.3, 0.1, 0.2)

(0.1, 0.0, 0.1)

(0.2 , 0.3 , 0.3 )

(x, 0.2, 0.3, 0.3)

b

(0.

) 0.1

(0.3, 0.1, 0.1) (w, 0.3, 0.1, 0.1) b

(u, 0.5, 0.3,b 0.1) (0.2, 0.3, 0.5)

, 0.3

1) (0.2, 0.3, 0.3)

b

.5, (0

3, 0 .

(0.2, 0.3, 0.5)

3, 0 .

) 0.2

(0.

(0.2, 0.3, 0.5)

(0.4, 0.3, 0.2)

(v, 0.2, 0.3, 0.5)

, 0.1

(0.2, 0.3, 0.3)

(0.5, 0.3, 0.1)

(u, 0.4, 0.1,b 0.2)

b

b

(0.1, 0.0, 0.0)

.4, (0

(v, 0.5, 0.3, 0.1)

b (0.3, 0.0, 0.1) (w, 0.3, 0.0, 0.1)

G = (QA, SB)

´ ´ 10. Neutrosophic soft rough graph Ǵ “ pǴ, Gq Figure Figure 10: Neutrosophic soft rough graph Ǵ = (Ǵ, G)

Definition 16. A graph G is called self complement, if G “ Ǵ, i.e.,

Definition 2.16. A graph G is called self complement, if G = Ǵ, i.e., (i)

@v P QA.

(i) ∀v ∈ QA.

´ ´ ´ “F ´ pvq “ T TQA ´ (v) QAp,vq , II´ QA pvq “ IQApvq , FQA pvq QApvq , TQA = TQA(v) QA (v) = IQA(v) , FQA (v) = FQA(v) , ´

´

´

TQA pvq “ TQApvq , I´QA pvq “ IQApvq , FQA pvq FQApvq . ´ (v) ´ “(v) = TQA(v) , IQA (v) = IQA(v) , FQA = FQA(v) .

TQA (ii)

@ v, u P QA.

(ii) ∀ v, u ∈ QA.

´

´

´

TSB= pvuq “ (vu), TSB pvuq,I ´ISB(vu) pvuq “ pvuq, FSB pvuq FSB pvuq, ´ “ ´ (vu) TSB = IISBSB (vu), FSB (vu) = FSB (vu), TSB SB ´ pvuq “ TSB pvuq, ISB ´ pvuq “ ISB pvuq, FSB ´ pvuq “ FSB pvuq.

TSB ´ (vu) = TSB (vu), ISB ´ (vu) = ISB (vu), FSB ´ (vu) = FSB (vu).

TSB

Definition 17. neutrosophic A neutrosophic soft rough graphgraph G is called strong neutrosophic rough graph ifsoft @uv rough P SB, graph if Definition 2.17. A soft rough G is called strong soft neutrosophic ∀uv ∈ SB,T pvuq “ mintT pvq, T puqu, I pvuq “ maxtI pvq, I puquq, F pvuq “ maxtF pvq, F puqu, SB

QA

QA

SB

QA

QA

SB

QA

QA

min{T (v),QA TQA ISBI(vu) = max{IQApvq, (v),IQA IQA (u)}), FSB =QAmax{F (v), FQA (u)} TSB (vu) =TSB pvuq “QA mintT pvq,(u)}, TQA puqu, puqu, FSB pvuq “(vu) maxtF pvq, FQAQA puqu.

SB pvuq “ maxtIQA TSB (vu) = min{TQA (v), TQA (u)}, ISB (vu) = max{IQA (v), IQA (u)}, FSB (vu) = max{FQA (v), FQA (u)}.

Example 9. Consider a graph G such that V “ tu, v, wu and E “ tuv, vw, wuu, as shown in Figure 11. Let QA be a neutrosophic soft rough set of V and let SB be a neutrosophic soft rough set of E defined in the Tables 9 and 10, respectively.

20

Table 9. Neutrosophic soft rough set on V. V

QA

QA

u v w

p0.8, 0.5, 0.2q p0.9, 0.5, 0.1q p0.7, 0.5, 0.1q

p0.7, 0.5, 0.2q p0.7, 0.5, 0.2q p0.7, 0.5, 0.2q

Table 10. Neutrosophic soft rough set on E. E

SB

SB

uv vw wu

p0.8, 0.5, 0.2q p0.7, 0.5, 0.1q p0.7, 0.5, 0.2q

p0.7, 0.5, 0.2q p0.7, 0.5, 0.2q p0.7, 0.5, 0.2q

Table 10: E uv vw wu

Axioms 2018, 7, 14

u(0.8,0.5,0.2)

b

v(0.7,0.5,0.2)

b

b

(0.8, 0.5, 0.2)

(0.7, 0.5, 0.2)

(0.7, 0.5, 0.2)

v(0.9,0.5,0.1)

Neutrosophic soft rough set on E SB SB (0.8, 0.5, 0.2) (0.7, 0.5, 0.2) (0.7, 0.5, 0.1) (0.7, 0.5, 0.2) (0.7, 0.5, 0.2) (0.7, 0.5, 0.2)

) , 0.2 , 0.5 (0.7

b

21 of 27

u(0.7,0.5,0.2) b

(0.7, 0.5, 0.2)

) , 0.2 , 0.5 (0.7

b

w(0.7,0.5,0.2)

w(0.7,0.5,0.2) G = (QA, SB)

G = (QA, SB)

Figure 11: 11. Strong neutrosophic soft rough graph Ggraph “ pQA,GSBq Figure Strong neutrosophic soft rough = (QA, SB)

Hence,GG=“(QA, pQA,SB) SBq isisa astrong neutrosophic soft rough Hence, strong neutrosophic soft graph. rough graph.

Definition 2.18.

A neutrosophic soft rough graph G isa called complete neutrosophic rough A neutrosophic soft rough graph G is called complete neutrosophic soft roughsoft graph if graph if Definition 18. ∀ @vu vu∈P QA, QA, (v),TTQA (u)}, = max{I (v),puqu, IQAF(u)}, F “ maxtF (vu) = max{FQA (v), FQA (u)}, TSB pvuq= “min{T mintTQA puqu, ISBIpvuq “ maxtI puqu, TSB(vu) QApvq, QA SB (vu) QAIQA QA pvq, SB pvuq SB QA pvq, FQA TSB (v),TTQA (u)}, = max{I (v),puqu, IQAF(u)}, FSB (vu) = (v), FQA (u)}.

QApvq, QA SB (vu) QAIQA QA TSB(vu) pvuq= “min{T mintTQA puqu, ISBIpvuq “ maxtI “ maxtF pvq, FQA puqu.

QA pvq, SB pvuq QAmax{F Remark Every complete neutrosophic soft rough graphneutrosophic is a strongsoft neutrosophic soft rough graph.

Remark2.2.

2. Every complete neutrosophic soft rough graph is a strong rough graph. However, But converse is not true. the the converse is not true.

Definition 2.19. A neutrosophic soft rough graph G is isolated, if ∀x, y ∈ QA. Definition 19. A neutrosophic soft rough graph G is isolated, if @x, y P QA.

TSB (vu) = 0, ISB (vu) = 0, FSB (vu) = 0, TSB (vu) = 0, ISB (vu) = 0, FSB (vu) = 0, TSB pvuq “ 0, ISB pvuq “ 0, FSB pvuq “ 0, TSB pvuq “ 0, ISB pvuq “ 0, FSB pvuq “ 0.

Theorem 2.1. The rejection of two neutrosophic soft rough graphs is a neutrosophic soft rough graph.

Theorem 1. The rejection of two neutrosophic soft rough graphs is a neutrosophic soft rough graph.

Proof. Let G1 = (G1 , G1 ) and G2 = (G2 , G2 ) be two NSRGs. Let G = G1 |G2 = (G1 |G2 , G1 |G2 ) be Proof.

Let G1of“GpG , G1 qGand G2 “ pG G22q = be (QA two 1NSRGs. Let1 |SB G “2G |G2 “ be the |QA2 , SB ) 1and G1pG |G1 |G (QA the rejection G21,|G 1 |G21q|QA 2 = 2 , SB1 |SB2 ).

1 1and 2 , where 2, G rejection G1 and , where G |QA , SB |SB q and G |G “ pQA |QA , SB |SB q.

We |G “ pQA We claim of that G =GG |G is an neutrosophic soft rough graph.

It is enough to show that SB1 |SB2 2 1 2 2 2 2 1 1 1 2 1 1 2 1 2 claim G2 “ |G2 is a neutrosophic graph.

It is enough show that SBFirst, SBthat areG1neutrosophic relationssoft onrough QA1 |QA QA , respectively. we show that and 1 |SB 2 and 1 |QA2 to 1 |SB2 and is aneutrosophic neutrosophicrelations relationon onQA QA SB SB11|SB |SB22 are QA1 |QA2 , respectively. First, we show that 1 |QA 2.

2 and 1 |QA SB1 |SB2 is a neutrosophic relation on QA1 |QA2 .

If v P QA1 , v1 v2 R SB2 , then “

21 ^ pTQA pv2 q ^ TQA pv2 qqq pTQA1 pvq 2 2

“

pTQA1 pvq ^ TQA2 pv2 qq ^ pTQA1 pvq ^ TQA2 pv2 qq

“

TpQA1 |QA2 q pv, v1 q ^ TpQA1 |QA2 q pv, v2 q

TpSB1 |SB2 q ppv, v1 qpv, v2 qq

“

TpQA1 |QA2 q pv, v1 q ^ TpQA1 |QA2 q pv, v2 q

Similarly, IpSB1 |SB2 q ppv, v1 qpv, v2 qq

“

IpQA1 |QA2 q pv, v1 q _ IpQA1 |QA2 q pv, v2 q

FpSB1 |SB2 q ppv, v1 qpv, v2 qq

“

FpQA1 |QA2 q pv, v1 q _ FpQA1 |QA2 q pv, v2 q.

TpSB1 |SB2 q ppv, v1 qpv, v2 qq

If v1 v2 R SB1 , v P QA2 , then TpSB1 |SB2 q ppv1 , vqpv2 , vqq

“

ppTQA1 pv1 q ^ TQA1 pv2 qq ^ TQA2 pvqq

“

ppTQA1 pv1 q ^ TQA2 pvqq ^ pTQA1 pv2 q ^ TQA2 pvqqq

“

TpQA1 |QA2 q pv1 , vq ^ TpQA1 |QA2 q pv2 , vq

TpSB1 |SB2 q ppv1 , vqpv2 , vqq

“

TpQA1 |QA2 q pv1 , vq ^ TpQA1 |QA2 q pv2 , vq

Similarly, IpSB1 |SB2 q ppv1 , vqpv2 , vqq

“

IpQA1 |QA2 q pv1 , vq _ IpQA1 |QA2 q pv2 , vq

FpSB1 |SB2 q ppv1 , vqpv2 , vqq

“

FpQA1 |QA2 q pv1 , vq _ FpQA1 |QA2 q pv2 , vq.

Axioms 2018, 7, 14

22 of 27

If v1 v2 R SB1 , u1 , u2 R SB2 , then TpSB1 |SB2 q ppv1 , u1 qpv2 , u2 qq

“

ppTQA1 pv1 q ^ TQA1 pv2 qq ^ pTQA2 pu1 q ^ TQA2 pu2 qqq

“

pTQA1 pv1 q ^ TQA2 pu1 qq ^ pTQA1 pv2 q ^ TQA2 pu2 qq

“

TpQA1 |QA2 q pv1 , u1 q ^ TpQA1 |QA2 q pv2 , u2 q

TpSB1 |SB2 q ppv1 , u1 qpv2 , u2 qq

“

TpQA1 |QA2 q pv1 , u1 q ^ TpQA1 |QA2 q pu1 , u2 q

Similarly, IpSB1 |SB2 q ppv1 , u1 qpv2 , u2 qq

“

IpQA1 |QA2 q pv1 , u1 q _ IpQA1 |QA2 q pu1 , u2 q

FpSB1 |SB2 q ppv1 , u1 qpv2 , u2 qq

“

FpQA1 |QA2 q pv1 , u1 q _ FpQA1 |QA2 q pu1 , u2 q.

Thus, SB1 |SB2 is a neutrosophic relation on QA1 |QA2 . Similarly, we can show that SB1 |SB2 is a neutrosophic relation on QA1 |QA2 . Hence, G is a neutrosophic soft rough graph.

Theorem 2. The Cartesian product of two NSRGs is a neutrosophic soft rough graph.

Proof. Let G1 “ pG1 , G1 q and G2 “ pG2 , G2 q be two NSRGs. Let G “ G1 ˙ G2 “ pG1 ˙ G2 , G1 ˙ G2 q be the Cartesian product of G1 and G2 , where G1 ˙ G2 “ pQA1 ˙ QA2 , SB1 ˙ SB2 q and G1 ˙ G2 “ pQA1 ˙ QA2 , SB1 ˙ SB2 q. We claim that G “ G1 ˙ G2 is a neutrosophic soft rough graph. It is enough to show that SB1 ˙ SB2 and SB1 ˙ SB2 are neutrosophic relations on QA1 ˙ QA2 and QA1 ˙ QA2 , respectively. We have to show that SB1 ˙ SB2 is a neutrosophic relation on QA1 ˙ QA2 .

If v P QA1 , v1 u1 P SB2 , then TpSB1 ˙SB2 q ppv, v1 qpv, u1 qq

“

TpQA1 q pvq ^ TpSB2 q pv1 u1 q

ď

TpQA1 q pvq ^ pTpQA2 q pv1 q ^ TpQA2 qpu1 qq

“

pTpQA1 q pvq ^ TpQA2 q pv1 qq ^ pTpQA1 q pvq ^ TpQA2 q pu1 qq

“

TpQA1 ˙QA2 q pv, v1 q ^ TpQA1 ˙QA2 q pv, u1 q

TpSB1 ˙SB2 q ppv, v1 qpv, u1 qq

ď

TpQA1 ˙QA2 q pv, v1 q ^ TpQA1 ˙QA2 q pv, u1 q

Similarly, IpSB1 ˙SB2 q ppv, v1 qpv, u1 qq

ď

IpQA1 ˙QA2 q pv, v1 q _ IpQA1 ˙QA2 q pv, u1 q

FpSB1 ˙SB2 q ppv, v1 qpv, u1 qq

ď

FpQA1 ˙QA2 q pv, v1 q _ FpQA1 ˙QA2 q pv, u1 q.

If v1 u1 P SB1 , z P QA2 , then TpSB1 ˙SB2 q ppv1 , zqpu1 , zqq

“

TpSB1 q pv1 u1 q ^ TpQA2 q pzq

ď

pTpQA1 qpv1 q^pQA1 q pu1 qq ^ TpQA2 q pzq

“

TpQA1 ˙QA2 q pv1 , zq ^ TpQA1 ˙QA2 q pu1 , zq

TpSB1 ˙SB2 q ppv1 , zqpu1 , zqq

ď

TpQA1 ˙QA2 q pv1 , zq ^ TpQA1 ˙QA2 q pu1 , zq

Similarly, IpSB1 ˙SB2 q ppv1 , zqpu1 , zqq

ď

IpQA1 ˙QA2 q pv1 , zq _ IpQA1 ˙QA2 q pu1 , zq

FpSB1 ˙SB2 q ppv1 , zqpu1 , zqq

ď

FpQA1 ˙QA2 q pv1 , zq _ FpQA1 ˙QA2 q pu1 , zq.

Therefore, SB1 ˙ SB2 is a neutrosophic relation on QA1 ˙ QA2 . Similarly, SB1 ˙ SB2 is a neutrosophic relation on QA1 ˙ QA2 . Hence, G is a neutrosophic rough graph.

Theorem 3. The cross product of two neutrosophic soft rough graphs is a neutrosophic soft rough graph.

Proof. Let G1 “ pG1 , G1 q and G2 “ pG2 , G2 q be two NSRGs. Let G “ G1 e G2 “ pG1 e G2 , G1 e G2 q be the cross product of G1 and G2 , where G1 e G2 “ pQA1 e QA2 , SB1 e SB2 q and G1 e G2 “ pQA1 e QA2 , SB1 e SB2 q. We claim that G “ G1 e G2 is a neutrosophic soft rough graph. It is enough to show that SB1 e SB2 and SB1 e SB2 are neutrosophic relations on QA1 e QA2 and QA1 e QA2 , respectively.

First, we show that SB1 e SB2 is a neutrosophic relation on QA1 e QA2 .

Axioms 2018, 7, 14

23 of 27

If v1 u1 P SB1 , v1 u2 P SB2 , then TpSB1 eSB2 q ppv1 , v1 qpu1 , u2 qq

“

TpSB1 q pv1 u1 q ^ TpSB2 q pv1 u2 q

ď

pTpQA1 q pv1 q ^ TpQA1 q pu1 q ^ pTpQA2 q pv1 q ^ TpQA2 q pu2 qq

“

pTpQA1 q pv1 q ^ TpQA2 q pv1 qq ^ pTpQA1 q pu1 q ^ TpQA2 q pu2 qq

“

TpQA1 eQA2 q pv1 , v1 q ^ TpQA1 eQA2 q pu1 , u2 q

TpSB1 eSB2 q ppv1 , v1 qpu1 , u2 qq

ď

TpQA1 eQA2 q pv1 , v1 q ^ TpQA1 eQA2 q pv, u2 q

Similarly, IpSB1 eSB2 q ppv1 , v1 qpu1 , u2 qq

ď

IpQA1 eQA2 q pv1 , v1 q _ IpQA1 eQA2 q pv, u2 q

FpSB1 eSB2 q ppv1 , v1 qpu1 , u2 qq

ď

FpQA1 eQA2 q pv1 , v1 q _ FpQA1 eQA2 q pv, u2 q.

Thus, SB1 e SB2 is a neutrosophic relation on QA1 e QA2 . Similarly, we can show that SB1 e SB2 is a neutrosophic relation on QA1 e QA2 . Hence, G is a neutrosophic soft rough graph.

3. Application In this section, we apply the concept of NSRSs to a decision-making problem. In recent times, the object recognition problem has gained considerable importance. The object recognition problem can be considered as a decision-making problem, in which final identification of objects is founded on a given set of information. A detailed description of the algorithm for the selection of most suitable objects based on an available set of alternatives is given, and purposed decision-making method can be used to calculate lower and upper approximation operators to progress deep concerns of the problem. The presented algorithms can be applied to avoid lengthy calculations when dealing with large number of objects. This method can be applied in various domains for multi-criteria selection of objects.

Selection of Most Suitable Generic Version of Brand Name Medicine In the pharmaceutical industry, different pharmaceutical companies develop, produce and discover pharmaceutical medicine (drugs) for use as medication. These pharmaceutical companies deals with “brand name medicine” and “generic medicine”. Brand name medicine and generic medicine are bioequivalent, and have a generic medicine rate and element of absorption. Brand name medicine and generic medicine have the same active ingredients, but the inactive ingredients may differ. The most important difference is cost. Generic medicine is less expensive as compared to brand name comparators. Usually, generic drug manufacturers face competition to produce cost less products. The product may possibly be slightly dissimilar in color, shape, or markings. The major difference is cost. We consider a brand name drug “u “ Loratadine” used for seasonal allergies medication. Consider V

“

tu1 “ Triamcinolone, u2 “ Cetirizine/Pseudoephedrine, u3 “ Pseudoephedrine, u4 “ loratadine/pseudoephedrine, u5 “ Fluticasoneu

is a set of generic versions of “Loratadine”. We want to select the most suitable generic version of Loratadine on the basis of parameters e1 “ Highly soluble, e2 “ Highly permeable, e3 “ Rapidly dissolving. M “ te1 , e2 , e3 u be a set of paraments. Let Q be a neutrosophic soft relation from V to parameter set M, and describe truth-membership, indeterminacy-membership and false-membership degrees of generic version medicine corresponding to the parameters as shown in Table 11.

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Table 11. Neutrosophic soft set pQ, Mq. Q

u1

u2

u3

u4

u5

e1 e2 e3

p0.4, 0.5, 0.6q p0.7, 0.3, 0.2q p0.6, 0.3, 0.4q

p0.5, 0.3, 0.6q p0.3, 0.4, 0.3q p0.7, 0.2, 0.3q

p0.7, 0.2, 0.3q p0.6, 0.5, 0.4q p0.7, 0.2, 0.4q

p0.5, 0.7, 0.5q p0.8, 0.4, 0.6q p0.8, 0.7, 0.6q

p0.6, 0.5, 0.4q p0.7, 0.8, 0.5q p0.7, 0.3, 0.5q

Suppose A “ tpe1 , 0.2, 0.4, 0.5q, pe2 , 0.5, 0.6, 0.4q, pe3 , 0.7, 0.5, 0.4qu is the most favorable object that is an NS on the parameter set M under consideration. Then, pQpAq, QpAqq is an NSRS in NSAS pV, M, Qq, where QpAq

“

tpu1 , 0.6, 0.5, 0.4q, pu2 , 0.7, 0.4, 0.4q, pu3 , 0.7, 0.4, 0.4q, pu4 , 0.7, 0.6, 0.5q, pu5 , 0.7, 0.5, 0.5qu,

QpAq

“

tpu1 , 0.5, 0.6, 0.4q, pu2 , 0.5, 0.6, 0.5q, pu3 , 0.3, 0.3, 0.5q, pu4 , 0.5, 0.6, 0.5q, pu5 , 0.4, 0.5, 0.5qu.

Let E “ tu1 v2 , u1 u3 , u4 u1 , u2 u3 , u5 u3 , u2 u4 , u2 u5 u Ď V́ and L “ te1 e3 , e2 e1 , e3 e2 u Ď Ḿ.

Then, a neutrosophic soft relation S on E (from L to E) can be defined as follows in Table 12: Table 12. Neutrosophic soft relation S. S

u1 u2

u1 u3

u4 u1

u2 u3

u5 u3

u2 u4

u2 u5

e1 e2 e2 e3 e1 e3

(0.3, 0.4 ,0.2) (0.5 ,0.4 ,0.1) (0.4,0.4,0.1)

p0.4, 0.4, 0.5q p0.4, 0.3, 0.2q p0.4, 0.2, 0.2q

p0.4, 0.4, 0.5q p0.4, 0.3, 0.2q p0.4, 0.2, 0.2q

p0.6, 0.3, 0.4q p0.3, 0.3, 0.2q p0.5, 0.3, 0.3q

p0.4, 0.2, 0.2q p0.6, 0.2, 0.4q p0.4, 0.2, 0.3q

p0.4, 0.4, 0.2q p0.3, 0.2, 0.1q p0.4, 0.3, 0.1q

p0.4, 0.3, 0.4q p0.3, 0.3, 0.2q p0.5, 0.3, 0.2q

Let B “ tpe1 e2 , 0.2, 0.4, 0.5q, pe2 e3 , 0.5, 0.4, 0.4q, pe1 e3 , 0.5, 0.2, 0.5qu be an NS on L that describes some relationship between the parameters under consideration; then, SB “ pSB, SBq is an NSRR, where SB

“

tpu1 u2 , 0.5, 0.4, 0.4q, pu1 u3 , 0.4, 0.2, 0.4q, pu4 u1 , 0.4, 0.2, 0.4q, pu2 u3 , 0.5, 0.3, 0.4q, pu5 u3 , 0.5, 0.2, 0.4q, pu2 u4 , 0.4, 0.3, 0.4q, pu2 u5 , 0.5, 0.3, 0.4qu,

SB

“

tpu1 u2 , 0.2, 0.4, 0.4qpu1 u3 , 0.5, 0.4, 0.4q, pu4 u1 , 0.5, 0.4, 0.4q, pu2 u3 , 0.4, 0.4, 0.5q, pu5 u3 , 0.2, 0.4, 0.4q, pu2 u4 , 0.2, 0.4, 0.4q, pu2 u5 , 0.4, 0.4, 0.5qu.

Thus, G “ pG, Gq is an NSRG as shown in Figure 12.

(0.4, 0.4, 0.5)

(u5 , 0.4, 0.5, 0.5)

(0.5, 0.4, 0.4)

(0.4, 0.2, 0.4)

5) 0.

4, . ,0 .4 (0

(u2 , 0.7, 0.4, 0.4)

(u4 , 0.7, 0.6, 0.5)

G = (QA, SB)

(u3 , 0.7, 0.4, 0.4)

.5, (0

(0 .4 ,0 .3 ,0 .4 ,)

(0.2, 0.4, 0.4, )

(u2 , 0.5, 0.6, 0.5)

(u3 , 0.3, 0.3, 0.5)

(0.2, 0.4, 0.4, )

(u4 , 0.5, 0.6, 0.5)

(0.4, 0.2, 0.4)

(u1 , 0.6, 0.5, 0.4)

(0.5 , 0.4 , 0.4 ) (0.2, 0.4, 0.4, )

4) 0.

4, . ,0 .5 (0

.4, ,0 0.3

(0 .5 ,0 .3 ,0 .4 )

)

(0.5, 0.2, 0.4)

(u1 , 0.5, 0.6, 0.4)

(u5 , 0.7, 0.5, 0.5)

G = (QA, SB)

Figure Neutrosophicsoft soft rough rough graph GqG) Figure 12:12.Neutrosophic graphGG“=pG, (G,

In [3], thethe sum of two neutrosophic numbers is defined.

In [3], sum of two neutrosophic numbers is defined.

Definition 3.1. [3] Let C and D be two single valued neutrosophic numbers, the sum of two single valued neutrosophic number is defined as follow: C ⊕ D =< TC + TD − TC × TD , IC × ID , FC × FD > .

(3.1)

The sum of UNSRS QA and the LNSRS QA and sum of LNSRR SB and the UNSRR SB are NSs QA ⊕ QA and SB ⊕ SB, respectively defined by

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Definition 20. [3] Let C and D be two single valued neutrosophic numbers, and the sum of two single valued neutrosophic number is defined as follows: C ‘ D “ă TC ` TD ´ TC ˆ TD , IC ˆ ID , FC ˆ FD ą .

(1)

Algorithm 1: Algorithm for selection of most suitable objects 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

Input the number of elements in vertex set V “ tu1 , u2 , . . . , un u.

Input the number of elements in parameter set M “ te1 , e2 , . . . , em u.

Input a neutrosophic soft relation Q from V to M. Input a neutrosophic set A on M. Compute neutrosophic soft rough vertex set QA “ pQA, QpAqq. Input the number of elements in edge set E “ tu1 u1 , u1 u2 , . . . , uk u1 u.

Input the number of elements in parameter set Ḿ “ te1 e1 , e1 e2 , . . . , el e1 u.

Input a neutrosophic soft relation S from V́ to Ḿ.

Input a neutrosophic set B on Ḿ.

Compute neutrosophic soft rough edge set SB “ pSB, SpBqq. Compute neutrosophic set α “ pTα pui q, Iα pui q, Fα pui qq, where Tα pui q

“

TQp Aq pui q ` TQp Aq pui q ´ TQp Aq pui q ˆ TQp Aq pui q,

Iα pui q

“

TQp Aq pui q ˆ TQp Aq pui q,

Fα pui q

“

FQp Aq pui q ˆ FQp Aq pui q.

Compute neutrosophic set β “ pTβ pui ui q, Iβ pui u j q, Fβ pui u j qq, where Tβ pui u j q

“

TSpBq pui u j q ` TSpBq pui u j q ´ TSpBq pui u j q ˆ TSpBq pui u j q,

Iβ pui u j q

“

TSpBq pui u j q ˆ TSpBq pui u j q,

Fβ pui u j q

“

FSpBq pui u j q ˆ FSpBq pui u j q.

Calculate the score values of each object ui , and the score function is defined as follows: ÿ

S̃pui q “

ui u j P E

Tα pu j q ` Iα pu j q ´ Fα pu j q .

3 ´ pTβ pui u j q ` Iβ pui u j q ´ Fβ pui u j qq

n

14.

The decision is Si if Si “ max S̃i .

15.

If i has more than one value, then any one of Si may be chosen.

i“1

The sum of UNSRS QA and the LNSRS QA and sum of LNSRR SB and the UNSRR SB are NSs QA ‘ QA and SB ‘ SB, respectively defined by α “ QA ‘ QA

“

tpu1 , 0.8, 0.3, 0.16q, pu2 , 0.85, 0.24, 0.2q, pu3 , 0.79, 0.2, 0.2q, pu4 , 0.85, 0.36, 0.25q, pu5 , 0.82, 0.25, 0.25qu,

β “ SB ‘ SB

“

tpu1 u2 , 0.6, 0.16, 0.16q, pu1 u3 , 0.7, 0.8, 0.16q, pu4 u1 , 0.7, 0.8, 0.16q, pu2 u3 , 0.7, 0.12, 0.2q, pu5 u3 , 0.6, 0.08, 0.16q, pu2 u4 , 0.52, 0.12, 0.16q, pu2 u5 , 0.7, 0.12, 0.2qu.

The score function S̃puk q defines for each generic version medicine ui P V, S̃pui q “

ÿ ui u j P E

Tα pu j q ` Iα pu j q ´ Fα pu j q 3 ´ pTβ pui u j q ` Iβ pui u j q ´ Fβ pui u j qq

(2)

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and uk with the larger score value uk “ max Spui q is the most suitable generic version medicine. i

By calculations, we have S̃pu1 q “ 0.88, S̃pu2 q “ 0.69, S̃pu3 q “ 0.26 S̃pu4 q “ 0.57, and S̃pu5 q “ 0.33.

(3)

Here, u1 is the optimal decision, and the most suitable generic version of “Loratadine” is “Triamcinolone”. We have used software MATLAB (version 7, MathWorks, Natick, MA, USA) for calculating the required results in the application. The algorithm is given in Algorithm 1. The algorithm of the program is general for any number of objects with respect to certain parameters.

4. Conclusions Rough set theory can be considered as an extension of classical set theory. Rough set theory is a very useful mathematical model to handle vagueness. NS theory, RS theory and SS theory are three useful distinguished approaches to deal with vagueness. NS and RS models are used to handle uncertainty, and combining these two models with another remarkable model of SSs gives more precise results for decision-making problems. In this paper, we have presented the notion of NSRGs and investigated some properties of NSRGs in detail. The notion of NSRGs can be utilized as a mathematical tool to deal with imprecise and unspecified information. In addition, a decision-making method based on NSRGs is proposed. This research work can be extended to (1) Rough bipolar neutrosophic soft sets; (2) Bipolar neutrosophic soft rough sets, (3) Interval-valued bipolar neutrosophic rough sets, and (4) Soft rough neutrosophic graphs.

Author Contributions: Muhammad Akram and Sundas Shahzadi conceived and designed the experiments; Hafsa M. Malik performed the experiments; Florentin Smarandache analyzed the data; Sundas Shahzadi and Hafsa M. Malik wrote the paper.

Conflicts of Interest: The authors declare no conflict of interest.

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