Achievable Single-Valued Neutrosophic Graphs in Wireless Sensor Networks

New Mathematics and Natural Computation Vol. 14, No. 2 (2018) 157{185 # .c World Scienti¯c Publishing Company DOI: 10.1142/S1793005718500114

Achievable Single-Valued Neutrosophic Graphs in Wireless Sensor Networks

Mohammad Hamidi* Department of Mathematics, Faculty of Mathematics Payame Noor University, Tehran, Iran m.hamidi@pnu.ac.ir

Arsham Borumand Saeid Department of Pure Mathematics Faculty of Mathematics and Computer Shahid Bahonar University of Kerman, Kerman, Iran arsham@uk.ac.ir

This paper considers wireless sensor (hyper) networks by single-valued neutrosophic (hyper) graphs. It tries to extend the notion of single-valued neutrosophic graphs to single-valued neutrosophic hypergraphs and it is derived single-valued neutrosophic graphs from single-valued neutrosophic hypergraphs via positive equivalence relation. We use single-valued neutrosophic hypergraphs and positive equivalence relation to create the sensor clusters and access to cluster heads. Finally, the concept of (extended) derivable single-valued neutrosophic graph is considered as the energy clustering of wireless sensor networks and is applied this concept as a tool in wireless sensor (hyper) networks.

Keywords: Single-valued neutrosophic (graphs) hypergraphs; positive equivalence relation; extendable single-valued neutrosophic graph; WSN. Mathematics Subject Classi¯cation (2010): 68R10, 05C15, 81Q30, 05C65

1. Introduction Neutrosophy, as a newly born science, is a branch of philosophy that studies the origin, nature and scope of neutralities, as well as their interactions with di®erent ideational spectra. It can be de¯ned as the incidence of the application of a law, an axiom, an idea, a conceptual accredited construction on an unclear, indeterminate phenomenon, contradictory to the purpose of making it intelligible. Neutrosophic set and neutrosophic logic are generalizations of the fuzzy set and respectively fuzzy logic (especially of intuitionistic fuzzy set and respectively intuitionistic fuzzy logic) are tools for publications on advanced studies in neutrosophy. In neutrosophic logic, *Corresponding author.

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a proposition has a degree of truth ðT Þ, indeterminacy ðIÞ and falsity ðF Þ, where T , I; F are standard or non-standard subsets of 0; 1 þ ½. In 1995, Smarandache talked for the ¯rst time about neutrosophy and in 1999 and 200511,12 he initiated the theory of neutrosophic set as a new mathematical tool for handling problems involving imprecise, indeterminacy, and inconsistent data. Alkhazaleh et al. generalized the concept of fuzzy soft set to neutrosophic soft set and they gave some applications of this concept in decision making and medical diagnosis.4 Smarandache13,14 have de¯ned four main categories of neutrosophic graphs, two based on literal indeterminacy ðIÞ, whose name were; I-edge neutrosophic graph and I-vertex neutrosophic graph, these concepts have been deeply studied and have gained popularity among the researchers due to their applications in real world problems.7,15 The two others graphs were based on ðt; i; fÞ components whose name was; The ðt; i; fÞ{Edge neutrosophic graph and the ðt; i; fÞ-vertex neutrosophic graph, these concepts are not developed at all. Later on, Broumi et al.5 introduced a third neutrosophic graph model. This model allows the attachment of truth-membership ðtÞ, indeterminacymembership ðiÞ and falsity-membership degrees ðfÞ both to vertices and edges, and investigated some of their properties. M. Akram et al. de¯ned the concepts of singlevalued neutrosophic hypergraph, line graph of single-valued neutrosophic hypergraph, dual single-valued neutrosophic hypergraph and transversal single-valued neutrosophic hypergraph.3 Wireless sensor networks ðWSNsÞ have gained worldwide attention in recent years, particularly with the proliferation of micro-electromechanical systems (MEMS) technology, which has facilitated the development of smart sensors. WSNs are used in numerous applications, such as environmental monitoring, habitat monitoring, prediction and detection of natural calamities, medical monitoring, and structural health monitoring. WSNs consist of tiny sensing devices that are spread over a large geographic area and can be used to collect and process environmental data such as temperature, humidity, light conditions, seismic activities, images of the environment, and so on.

Regarding these points, this paper aims to generalize the notion of single-valued neutrosophic graphs by considering the notion of positive equivalence relation and trying to de¯ne the concept of derivable single-valued neutrosophic graphs. The relationships between derivable single-valued neutrosophic graphs and single-valued neutrosophic hypergraphs are considered as a natural question. The quotient of single-valued neutrosophic hypergraphs via equivalence relations is the main motivation of this research. Moreover, by using positive equivalence relations, we de¯ne a well-de¯ned operation on single-valued neutrosophic hypergraphs that the quotient of any single-valued neutrosophic hypergraphs via this relation is a single-valued neutrosophic graph. We use single-valued neutrosophic hypergraphs to represent wireless sensor hypernetworks. By considering the concept of the wireless sensor networks, the use of wireless sensor hypernetworks appears to be a necessity for exploring these systems and representation their relationships. We have introduced several valuable measures as truth-membership, indeterminacy and falsity-membership

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values for studying wireless sensor hypernetworks, such as node and hypergraph centralities as well as clustering coe±cients for both hypernetworks and networks.

Clustering is one of the basic approaches for designing energy-e±cient, robust and highly scalable distributed sensor networks. A sensor network reduces the communication overhead by clustering, and decreases the energy consumption and the interference among the sensor nodes, so we via the concept of single-valued neutrosophic (hyper) graphs and equivalence relations considered the wireless sensor hypernetworks.

2. Preliminaries In this section, we recall some de¯nitions and results are indispensable to our research paper.

De¯nition 2.1 (Ref. 6). Let G ¼ fx1 ; x2 ; . . . ; xn g be a ¯nite set. A hypergraph on G is a family H ¼ ðE1 ; E2 ; . . . ; Em Þ ¼ ðG; fEi g m i¼1 Þ of subsets of G such that (i) for all 1 i m; Ei 6¼ ;; S (ii) m i¼1 Ei ¼ G. A simple hypergraph (Sperner family) is a hypergraph H ¼ ðE1 ; E2 ; . . . ; Em Þ such that (iii) Ei Ej ) i ¼ j.

The elements x1 ; x2 ; . . . ; xn of G are called vertices, and the sets E1 ; E2 ; . . . ; Em are the edges (hyperedges) of the hypergraph. For any 1 k m if jEk j 2, then Ek is represented by a solid line surrounding its vertices, if jEk j ¼ 1 by a cycle on the element (loop). If for all 1 k m jEk j ¼ 2, the hypergraph becomes an ordinary (undirected) graph.

De¯nition 2.2 (Ref. 9). Let H ¼ ðG; fEx gx2G Þ be a hypergraph. Then H ¼ ðG; fEx gx2G Þ is called a complete hypergraph, if for any x; y 2 G there is a hyperedge E such that fx; yg E and is shown a complete hypergraph with n elements by K n .

Let H ¼ ðG; fEi g nþ1 i¼1 Þ be a complete hypergraph. (i) H ¼ ðG; fEi g nþ1 i¼1 Þ is called a joint complete hypergraph, if for any 1 i n, jEi j ¼ i; Ei Eiþ1 and jEnþ1 j ¼ n; (ii) H ¼ ðG; fEi g nþ1 i¼1 Þ is called a discrete complete hypergraph, if for any 1 i 6¼ j n; jEi j ¼ jEj j; Ei \ Ej ¼ ; and jEnþ1 j ¼ n; De¯nition 2.3 (Ref. 10). Let H ¼ ðG; fEi g ni¼1 Þ be a hypergraph. De¯ne a binary relation on G as follows: 1 ¼ fðx; xÞ j x 2 Gg and for every integer k > 1, x k y , 9 E is , such that fx; yg E is , where k ¼ jE is j ¼ minfjEt j; x; y 2 Et g and for all 1 i; j n; there is no Ei 6¼ E is , or Ej 6¼ E is , such that x 2 Ei ; y 2 Ej and S jEi j < k; jEj j < k. Obviously ¼ k 1 k is a re°exive and symmetric relation on G. Let be the transitive closure of (the smallest transitive relation such that contains ).

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Theorem 2.4 (Ref. 10). Let H ¼ ðG; fEx gx2G Þ be a hypergraph, N ¼ N [ f0g and ¼ . Then for any i 2 N there exists an operation i on G= such that H= ¼ ðG= ; i Þ is a graph.

De¯nition 2.5 (Ref. 16). Let X be a set. A single valued neutrosophic set A in X ðSVN{S AÞ is a function A : X ! ½0; 1 ½0; 1 ½0; 1 with the form A ¼ fðx; TA ðxÞ; IA ðxÞ; FA ðxÞÞ j x 2 Xg where the functions TA ; IA ; FA de¯ne respectively the truth-membership function, an indeterminacy-membership function, and a falsity-membership function of the element x 2 X to the set A such that 0 TA ðxÞ þ IA ðxÞ þ FA ðxÞ 3. Moreover, SuppðAÞ ¼ fx j TA ðxÞ 6¼ 0; IA ðxÞ 6¼ 0; FA ðxÞ 6¼ 0g is a crisp set.

De¯nition 2.6 (Ref. 5). A single valued neutrosophic graph (SVN-G) is de¯ned to be a form G ¼ ðV ; E; A; BÞ where (i) V ¼ fv1 ; v2 ; . . . ; vn g, TA ; IA ; FA : V ! ½0; 1 denote the degree of membership, degree of indeterminacy and non–membership of the element vi 2 V ; respectively, and for every 1 i n; we have 0 TA ðvi Þ þ IA ðvi Þ þ FA ðvi Þ 3. (ii) E V V , TB ; IB ; FB : E ! ½0; 1 are called degree of truth-membership, indeterminacy-membership and falsity-membership of the edge ðvi ; vj Þ 2 E respectively, such that for any 1 i; j n; we have TB ðvi ; vj Þ minfTA ðvi Þ; TA ðvj Þg; IB ðvi ; vj Þ maxfIA ðvi Þ; IA ðvj Þg, FB ðvi ; vj Þ maxfFA ðvi Þ; FA ðvj Þg and 0 TB ðvi ; vj Þ þ IB ðvi ; vj Þ þ FB ðvi ; vj Þ 3. Also A is called the single valued neutrosophic vertex set of V and B is called the single valued neutrosophic edge set of E. De¯nition 2.7 (Ref. 3). (i) A single valued neutrosophic hypergraph (SVN–HG) is de¯ned to be a pair H ¼ ðV ; fEi g m i¼1 Þ, where V ¼ fv1 ; v2 ; . . . ; vn g is a ¯nite set of vertices and fEi ¼ fðvj ; TEi ðvj Þ; IEi ðvj Þ; FEi ðvj ÞÞgg m i¼1 is a ¯nite family of non-trivial neutrosophic S m subsets of the vertex V such that V ¼ m i¼1 suppðEi Þ. Also fEi g i¼1 is called the family of single valued neutrosophic hyperedges of H and V is the crisp vertex set of H. (ii) Let 1 ; ; 1, then A ð ; ; Þ ¼ fx 2 X j TA ðxÞ ; IA ðxÞ ; FA ðxÞ g is called ð ; ; Þ-level subset of A. 3. (Regular) Single-Valued Neutrosophic Hypergraphs (Graphs) (SVN{HG) In this section, we introduce a concept of regular single-valued neutrosophic graph and construct quotient single-valued neutrosophic hypergraphs, via equivalence relations.

Let ðH; fEi g ni¼1 Þ be a hypergraph, 1 i; j n and k 2 N. Then H is called a partitioned hypergraph, if P ¼ fE1 ; E2 ; . . . ; En g is a partition set of H. We will

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denote the set of partitioned hypergraphs with jPj ¼ k on H that jEi j ¼ jEj j, by ðkÞ

P h ðHÞ and the set of all partitioned hypergraphs on H, by P h ðHÞ.

De¯nition 3.1. Let G ¼ ðV ; E; A; BÞ be a single-valued neutrosophic graph. Then G ¼ ðV ; E; A; BÞ is called (i) a weak single-valued neutrosophic graph, if suppðAÞ ¼ V ; (ii) a regular single-valued neutrosophic graph, if is weak and for any vi ; vj 2 V have TB ðvi ; vj Þ ¼ minfTA ðvi Þ; TA ðvj Þg; IB ðvi ; vj Þ ¼ maxfIA ðvi Þ; IA ðvj Þg and FB ðvi ; vj Þ ¼ maxfFA ðvi Þ; FA ðvj Þg.

Proposition 3.2. Let V ¼ fa1 ; a2 ; . . . ; an g. Consider the complete graph Kn and de¯ne A : V ! ½0; 1 by TA ðai Þ ¼ 1=i; IA ðai Þ ¼ 1=ði þ 1Þ; FA ðai Þ ¼ 1=ði þ 2Þ and ðiÞ B : V V ! ½0; 1 by TB ðai ; aj Þ ¼ TA ðai Þ TA ðaj Þ, IB ðai ; aj Þ ¼ FB ðai ; aj Þ ¼ IA ðai Þ þ TA ðaj Þ. It is clear that G ¼ ðV ; E; A; BÞ is a single-valued neutrosophic complete graph and since suppðAÞ ¼ V , we get that it is a weak single-valued neutrosophic complete graph. ðiiÞ B : V V ! ½0; 1 by TB ðai ; aj Þ ¼ jTA ðai Þ þ TA ðaj Þj=2 jTA ðai Þ TA ðaj Þj=2; IB ðai ; aj Þ ¼ jIA ðai Þ þ IA ðaj Þj=2 þ jIA ðai Þ IA ðaj Þj=2 and FB ðai ; aj Þ ¼ jFA ðai Þ þ FA ðaj Þj=2 þ jFA ðai Þ FA ðaj Þj=2. It is clear that G ¼ ðV ; E; A; BÞ is a regular single-valued neutrosophic complete graph.

Corollary 3.3. Any ¯nite set can be a ðregularÞweak single-valued neutrosophic complete graph.

Proof. Let G be a ¯nite set and R be an equivalence relation on G. Then consider, H ¼ ðG; fRðxÞ RðyÞgx;y2G Þ, whence it is a complete graph. By Proposition 3.2, is obtained.

Lemma 3.4. Let X be a ¯nite set and A ¼ fðx; TA ðxÞ; IA ðxÞ; FA ðxÞÞ j x 2 Xg be a single-valued neutrosophic set in X. If R is an equivalence relation on X, then A=R ¼ fðRðxÞ; TRðAÞ ðRðxÞÞ; IRðAÞ ðRðxÞÞ; FRðAÞ ðRðxÞÞ j x 2 Xg is a single-valued neutrosophic set, where TRðAÞ ðRðxÞÞ ¼ ^t R x TA ðtÞ; IRðAÞ ðRðxÞÞ ¼ _t R x IA ðtÞ and FRðAÞ ðRðxÞÞ ¼ _t R x FA ðtÞ.

Proof. Let X ¼ fx1 ; x2 ; . . . ; xn g and P ¼ fRðx1 Þ; Rðx2 Þ; . . . ; Rðxk Þg be a partition of X, where k n. Since for any xi 2 X; TA ðxi Þ 1; IA ðxi Þ 1 and FA ðxi Þ 1, we get that ^t R xi TA ðtÞ 1; _t R xi IA ðtÞ 1 and _t R xi FA ðtÞ 1. Hence for any 1 i k, 0 ^t R xi TA ðtÞ þ _t R xi IA ðtÞ þ _t R xi FA ðtÞ 3 and so RðAÞ ¼ fðRðxi Þ; ^t R xi TA ðtÞ; _t R xi IA ðtÞ; _t R xi FA ðtÞÞg ki¼1 is a single-valued neutrosophic set in X=R. Theorem 3.5. Let V ¼ fv1 , v2 ;...;vn g and H ¼ ðV , fðvj ;TEi ðvj Þ; IEi ðvj Þ;FEi ðvj ÞÞg m i¼1 Þ be a single-valued neutrosophic hypergraph. If R is an equivalence relation on H, then H=R ¼ ðRðV Þ; fRðvj Þ; TRðEi Þ ðRðvj ÞÞ; IRðEi Þ ðRðvj ÞÞ; FRðEi Þ ðRðvj ÞÞg m i¼1 Þ is a partitioned single-valued neutrosophic hypergraph.

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Proof. By Lemma 3.4, fRðvj Þ; TRðEi Þ ðRðvj ÞÞ; IRðEi Þ ðRðvj ÞÞ; FRðEi Þ ðRðvj ÞÞg m i¼1 is a S ¯nite family of single-valued neutrosophic subsets of V =R. Since V ¼ m suppðE i Þ, i¼1 Sm Sm we get that i¼1 suppðRðEi ÞÞ ¼ Rð i¼1 suppðEi ÞÞ ¼ RðV Þ. It follows that H=R ¼ ðRðV Þ; fRðvj Þ; TRðEi Þ ðRðvj ÞÞ; IRðEi Þ ðRðvj ÞÞ; FRðEi Þ ðRðvj ÞÞg m i¼1 Þ is a single-valued neutrosophic hypergraph. Since R is an equivalence relation on V , for any x 6¼ y 2 V we get that RðxÞ \ RðyÞ ¼ ; and so it is a partitioned single-valued neutrosophic hypergraph.

Example 3.6. Consider a joint complete single-valued neutrosophic hypergraph H ¼ ðV ; fEi g ni¼1 Þ, where V ¼ fa1 ; a2 ; . . . ; an g and for any 1 i n; Ei ¼ fðai ; i=10 n ; ði þ 1Þ=10 n Þ; ði þ 2Þ=10 n Þg. Clearly R ¼ fðai ; ai Þ; ðar ; as Þ j r þ s ¼ n þ 1; 1 i ng is an equivalence relation on V and so we obtain V =R ¼ fRða1 Þ; Rða2 Þ; Rða3 Þ; . . . ; Rðaðn=2Þ 1 Þ; Rðan=2 Þg. It follows that H=R ¼ ðfRða1 Þ; Rða2 Þ; Rða3 Þ; . . . ; Rðaðn=2Þ 1 Þ; Rðan=2 Þg; fðRðai Þ; i=10 n ; ði þ 1Þ=10 n ; ði þ 2Þ=10 n Þ; ðRðan iþ1 Þ; n=2

ðn i þ 1Þ=10 n ; ðn i þ 2Þ=10 n ; ðn i þ 3Þ=10 n Þg i¼1 Þ: Computation shows that H=R is a partitioned single-valued neutrosophic hypergraph.

4. Derivable Regular Single-Valued Neutrosophic Graphs In this section, we introduce the concept of derivable single-valued neutrosophic graphs via the equivalence relation on single-valued neutrosophic hypergraphs. It is shown that any single-valued neutrosophic graph is not necessarily a derivable singlevalued neutrosophic graph and it is proved under some conditions. Furthermore, it can show that some regular trees and all regular single-valued neutrosophic complete bigraphs are derivable single-valued neutrosophic graph and regular single-valued neutrosophic complete graph are self derivable single-valued neutrosophic graphs.

De¯nition 4.1. A single-valued neutrosophic graph G ¼ ðV ; E; A; BÞ is said to be: (i) a derivable single-valued neutrosophic graph if there exists a nontrivial singlevalued neutrosophic hypergraph ðH; fEk g nk¼1 Þ such that ðH; fEk g nk¼1 Þ= ffi G ¼ ðV ; E; A; BÞ and H is called an associated single-valued neutrosophic hypergraph with single-valued neutrosophic graph G. In other words, it is equal to the quotient of nontrivial single-valued neutrosophic hypergraph on up to isomorphic; (ii) a self derivable single-valued neutrosophic graph, if it is a derivable single-valued neutrosophic graph by itself.

Theorem 4.2. Let H ¼ ðV ; fEi g m i¼1 Þ be a single-valued neutrosophic hypergraph, j 2 N ¼ N [ f0g and ¼ . Then there exists an operation \ j " on H= such that ðH= ; j Þ is a regular single-valued neutrosophic graph.

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Proof. By Theorem 3.5, H= ¼ ð ðV Þ, f ðvj Þ, T ðEi Þ ð ðvj ÞÞ, I ðEi Þ ð ðvj ÞÞ, F ðEi Þ ð ðvj ÞÞg m i¼1 Þ is a partitioned single-valued neutrosophic hypergraph, where V W TEi ðtÞ; I ðEi Þ ð ðxÞÞ ¼ IEi ðtÞ and T ðEi Þ ð ðxÞÞ ¼ x t2X

F ðEi Þ ð ðxÞÞ ¼

W

x t2X

x t2X

FEi ðtÞ:

For any ðxÞ ¼ ððx; TEi ðxÞ; IEi ðxÞ; FEi ðxÞÞÞ and ðyÞ ¼ ððy; TEi ðyÞ; IEi ðyÞ; FEi ðyÞÞ 2 H= , de¯ne an operation \ j " on H= by ( c ðxÞ; ðyÞ if jj ðxÞj j ðyÞjj ¼ j; ðxÞ j ðyÞ ¼ b ; otherwise; c where for any x; y 2 G; ð ðxÞ; ðyÞÞ is represented as an ordinary (simple) edge and b ;¼ d ðx Þ means that there is not edge. It is easy to see that H= ¼ ð ðV Þ; f ðvj Þ; T ðEi Þ ð ðvj ÞÞ; I ðEi Þ ð ðvj ÞÞ; F ðEi Þ ð ðvj ÞÞg m i¼1 ; j Þ is a graph. Now, de¯ne T ðEi Þ ; I ðEi Þ ; F ðEi Þ : ðV Þ ðV Þ ! ½0; 1 by T ðEi Þ ð ðxÞ, ðyÞÞ ¼ ^a x;b y ðT ðEi Þ ðaÞ ^ T ðEi Þ ðbÞÞ; I ðEi Þ ð ðxÞ; ðyÞÞ ¼ _a x;b y ðI ðEi Þ ðaÞ _ I ðEi Þ ðbÞÞ and F ðEi Þ ð ðxÞ; ðyÞÞ ¼ _a x;

b y ðF ðEi Þ ðaÞ

_ F ðEi Þ ðbÞÞ. It is clear to see that T ðEi Þ

ð ðxÞ; ðyÞÞ ðT ðEi Þ ð ðxÞÞ ^ T ðEi Þ ð ðyÞÞÞ; I ðEi Þ ð ðxÞ; ðyÞÞ ðI ðEi Þ ð ðxÞÞ_ I ðEi Þ ð ðyÞÞÞ and F ðEi Þ ð ðxÞ; ðyÞÞ ðF ðEi Þ ð ðxÞÞ _ F ðEi Þ ð ðyÞÞÞ. Hence, H= ¼ ð ðV Þ; f ðvj Þ; T ðEi Þ ð ðvj ÞÞ; I ðEi Þ ð ðvj ÞÞ; F ðEi Þ ð ðvj ÞÞg m i¼1 ; Þ is a single-valued neutrosophic graph.

Example 4.3. Let H ¼ ðfa; b; c; d; e; f; gg; fE1 ; E2 ; E3 ; E4 gÞ be a single-valued neutrosophic hypergraph in Fig. 1. Since

Fig. 1.

(d, 0.6, 0.5, 0.4)

(c, 0.4, 0.5, 0.6)

(e, 0.7, 0.8, 0.9)

(f, 0.9, 0.8, 0.7)

(g, 0.1, 0.3, 0.5)

E 1s ¼ fða; 0:1; 0:2; 0:3Þ; ðb; 0:3; 0:2; 0:1Þg; E 2s ¼ fðc; 0:4; 0:5; 0:6Þ; ðd; 0:6; 0:5; 0:4Þg; E 3s ¼ fðe; 0:7; 0:8; 0:9Þ; ðf; 0:9; 0:8; 0:7Þg and E 4s ¼ fðg; 0:1; 0:3; 0:5Þg;

SVN-HG.

(a, 0.1, 0.2, 0.3) (b, 0.3, 0.2, 0.1)

M. Hamidi & A. Borumand Saeid

0.

9)

(0.4, 0.8, 0.9)

0.

8,

(η(g), 0.1, 0.3, 0.5) •

• (η(c), 0.4, 0.5, 0.6)

0.

6)

, .1 (0

(η (e ), 0

.7 ,

•

9) 0.

(0 .1 ,

8, 0.

0.

5,

164

• (η(a), 0.1, 0.2, 0.3)

Fig. 2. Derivable SVN-G G= for i ¼ 0.

by Theorem 4.2, we get that ¼ and ðða; 0:1; 0:2; 0:3ÞÞ ¼ ððb; 0:3; 0:2; 0:1ÞÞ ¼ fða; 0:1; 0:2; 0:3Þ; ðb; 0:1; 0:2; 0:3Þg; ððc; 0:4; 0:5; 0:6ÞÞ ¼ ððd; 0:6; 0:5; 0:4ÞÞ ¼ fðc; 0:4; 0:5; 0:6Þ; ðd; 0:4; 0:5; 0:6Þg; and

ððe; 0:7; 0:8; 0:9ÞÞ ¼ ððf; 0:9; 0:8; 0:7ÞÞ ¼ fðe; 0:7; 0:8; 0:9Þ; ðf; 0:7; 0:8; 0:9Þg; ððg; 0:1; 0:3; 0:5ÞÞ ¼ fðg; 0:1; 0:3; 0:5Þg:

Now, for i ¼ 0, we obtain the regular single-valued neutrosophic graph in Fig. 2.

G= is a regular unconnected single-valued neutrosophic graph with 4 vertices and 3 edges. For i ¼ 1, we obtain G= ffi K1;3 as Fig. 3. G= is a connected regular singlevalued neutrosophic graph with 4 vertices and 3 edges. Moreover, for any i 2 graph G= is isomorphic to null single-valued neutrosophic graph K 4 (Fig. 4).

(0 .1 ,

0.

5,

)

(0 .1 ,

, 0.5 , 0.3 (0.1

(η(g), 0.1, 0.3, 0.5)

0.

6)

•

0.

8,

• • (η(a), 0.1, 0.2, 0.3) (η(c), 0.4, 0.5, 0.6)

0.

9) • (η(e), 0.7, 0.8, 0.9)

Fig. 3. Derivable SVN-G G= for i ¼ 1.

(η(a), 0.1, 0.2, 0.3) •

(η(g), 0.1, 0.3, 0.5)

• (η(c), 0.4, 0.5, 0.6)

• (η(e), 0.7, 0.8, 0.9)

•

Fig. 4. Derivable SVN-G G= for i 2.

Fig. 5.

165

(a1 , 0.2, 0.4, 0.6)

(a2 , 0.4, 0.6, 0.6)

(a3 , 0.3, 0.2, 0.4)

(a4 , 0.8, 0.9, 0.1)

(a5 , 0.1, 0.9, 0.9)

Achievable Single-Valued Neutrosophic Graphs in Wireless Sensor Networks

SVN-HG.

Example 4.4. Let V ¼ fa1 ; a2 ; a3 ; a4 ; a5 g. Then consider the single-valued neutrosophic hypergraph H ¼ ðV ; E1 ; E2 ; E3 ; E4 ; E5 Þ in Fig. 5. Clearly ¼ , E as1 ¼ fða1 ; 0:2; 0:4; 0:6Þg; E as4 ¼ fða4 ; 0:8; 0:9; 0:1Þg

E as2 ¼ fða2 ; 0:4; 0:6; 0:6Þg;

E as3 ¼ fða3 ; 0:3; 0:2; 0:4Þg;

E as5 ¼ fða5 ; 0:1; 0:9; 0:9Þg;

and

thus by Theorem 4.2, we obtain ðða1 ; 0:2; 0:4; 0:6ÞÞ ¼ fða1 ; 0:2; 0:4; 0:6Þg; ðða3 ; 0:3; 0:2; 0:4ÞÞ ¼ fða3 ; 0:3; 0:2; 0:4Þg; and

ðða2 ; 0:4; 0:6; 0:6ÞÞ ¼ fða2 ; 0:4; 0:6; 0:6Þg; ðða4 ; 0:8; 0:9; 0:1ÞÞ ¼ fða4 ; 0:8; 0:9; 0:1Þg

ðða5 ; 0:1; 0:9; 0:9ÞÞ ¼ fða5 ; 0:1; 0:9; 0:9Þg:

Now, for i ¼ 0, H= ¼ ð ðV Þ, fð ðai Þ; T ðEi Þ ð ðai ÞÞ; I ðEi Þ ð ðai ÞÞ, F ðEi Þ ð ðai ÞÞÞg 5i¼1 and we obtain the regular single-valued neutrosophic graph in Fig. 6.

(0.8, 0.9, 0.1) η(a4 ) •

•η(a2 ) (0.4, 0.6, 0.6)

(0.1, 0.9, 0.9)

, .4 (0

9, 0.

6) 0.

(0 .1 ,

0.

9,

0.

9)

(0.3, 0.9, 0.4) Fig. 6. Derived cycle SVN-G K5 .

(0.3, 0.6, 0.6)

(0.1, 0.9, 0.9)

(0.1, 0.9, 0.9) η(a5 ) •

6) 0.

(0 .1 ,

6, 0.

0.

9,

, .2 (0

0.

9)

η(a1 )(0.2, 0.4, 0.6) •

• η(a3 )(0.3, 0.2, 0.4)

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Theorem 4.5. Let G ¼ ðV ; E; A; BÞ be a derivable single-valued neutrosophic graph by a single-valued neutrosophic hypergraph H ¼ ðV ; E ¼ fvj ; TEk ðvj Þ; IEk ðvj Þ; FEk ðvj ÞÞg m k¼1 Þ. Then ðiÞ jV = j ¼ jV j and jEj ¼ jEj; ðiiÞ jV j jV j; ðiiiÞ if G ¼ ðV ; E; A; BÞ is a connected single-valued neutrosophic graph and jV j ¼ jV j, then i ¼ 0 ; ðivÞ for any 1 j m, T ðEi Þ ðvj Þ TEi ðvj Þ, I ðEi Þ ðvj Þ IEi ðvj Þ and F ðEi Þ ðvj Þ FEi ðvj Þ.

Proof. (iii) Let jV j ¼ jV j. ððy; TEi ðyÞ; IEi ðyÞ; FEi ðyÞÞÞ. neutrosophic graph, we get j ððy; TEi ðyÞ; IEi ðyÞ; FEi ðyÞÞÞj

Then for any x; y 2 V ; ððx; TEi ðxÞ; IEi ðxÞ; FEi ðxÞÞÞ 6¼ Since G ¼ ðV ; E; A; BÞ is a connected single-valued that for any x; y 2 V ; j ððx; TEi ðxÞ; IEi ðxÞ; FEi ðxÞÞÞj ¼ and so i ¼ 0 .

Example 4.6. Let V ¼ fa1 ; a2 ; . . . ; an g and i ¼ 0. Consider the discrete complete hypergraph in Fig. 7. Then it is easy to see that K n ¼ ðV ; fEk ¼ fðvk ; TEk ðvj Þ; IEk ðvj Þ; FEk ðvj ÞÞgg nþ1 hyperk¼1 Þ is a discrete complete single-valued neutrosophic S graph, where for any 1 k n; Ek ¼ fðak ; 1=k; 1=k 2 ; 1=k 3 Þg and Enþ1 ¼ nk¼1 Ek .

Clearly for any 1 k n; ðak Þ ¼ E ask ¼ fðak ; 1=k; 1=k 2 ; 1=k 3 Þg. Hence K n = ¼ f ðak Þ j 1 k ng and so K n = ffi Kn . For any 1 k n; we get T ðEi Þ ð ðak Þ; ðakþ1 ÞÞ ¼ ð1=kÞ ^ 1=ðk þ 1Þ ¼ 1=ðk þ 1Þ T ðEi Þ ð ðak ÞÞ ^ T ðEi Þ ð ðakþ1 ÞÞ. Therefore, for any n 2 N; Kn is a derivable single-valued neutrosophic graph.

Theorem 4.7. Let G ¼ ðV ; E; A; BÞ be a connected single-valued neutrosophic graph. G is a self derivable single-valued neutrosophic graph if and only if G is a single-valued neutrosophic complete graph.

Proof. Let V ¼ fa1 ; a2 ; . . . ; an g. If G is a single-valued neutrosophic complete graph, then by Example 4.6, G is a self derivable single-valued neutrosophic graph.

Conversely, let G be a self derivable single-valued neutrosophic graph. Then by c Theorem 4.5, we get ¼ and so for any x; y 2 G; ðxÞ ðyÞ ¼ ðxÞ; ðyÞ. Thus, i

0

i

G ¼ ðV ; E; A; BÞ is a derivable single-valued neutrosophic complete graph.

a1

a2

Fig. 7.

a3

a4 ...... an−1 an

Joint complete SVN-HG K n .

Achievable Single-Valued Neutrosophic Graphs in Wireless Sensor Networks

167

4.1. Derivable cycle single-valued neutrosophic graph In this section, we consider derivable cycle single-valued neutrosophic graph and show that Cn is a derivable cycle single-valued neutrosophic graph if and only if n 2 f3; 4g.

Example 4.8. Consider the cycle single-valued neutrosophic graph C4 in Fig. 8.

Now introduce the single-valued neutrosophic hypergraph H in Fig. 9. Clearly ¼ , ðða1 ; 0:9; 0:8; 0:1ÞÞ ¼ fða1 ; 0:9; 0:8; 0:1Þg; ðða2 ; 0:8; 0:1; 0:1ÞÞ ¼ fða2 ; 0:2; 0:1; 0:3Þ; ðb1 ; 0:2; 0:1; 0:3Þg; ðða3 ; 0:3; 0:2; 0:1ÞÞ ¼ fða3 ; 0:3; 0:2; 0:1Þg and ðða4 ; 0:4; 0:1; 0:1ÞÞ ¼ fða4 ; 0:4; 0:3; 0:2Þ; ðb2 ; 0:4; 0:3; 0:2Þg:

By Theorem 4.2 and for i ¼ 1; we obtain that H= ffi C4 .

Lemma 4.9. ðC5 ; A; BÞ is not a derivable single-valued neutrosophic graph.

Proof. Consider the cycle single-valued neutrosophic graph C5 in Fig. 10, where for any 1 j 5 we have 0 Tj þ Ij þ Fj 3, j ¼ ðTjjþ1 ; Ijjþ1 ; Fjjþ1 Þ and (0.2, 0.8, 0.3)

• (a1 , 0.9, 0.8, 0.1)

(0.2, 0.2, 0.3)

(0.4, 0.8, 0.2)

(a2 , 0.2, 0.1, 0.3) •

(0.3, 0.3, 0.2)

(a3 , 0.3, 0.2, 0.1) •

• (a4 , 0.4, 0.3, 0.2)

Fig. 9.

(b2 , 0.5, 0.3, 0.2)

(a4 , 0.4, 0.1, 0.1)

(a1 , 0.9, 0.8, 0.1)

(a3 , 0.3, 0.2, 0.1)

Fig. 8. Derived cycle SVN-G C4 .

(a2 , 0.8, 0.1, 0.1)

SVN-HG.

(b1 , 0.2, 0, 0.3)

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(a1 , T1 , I1 , F1 ) • λ5 λ1 • (a2 , T2 , I2 , F2 ) (a5 , T5 , I5 , F5 ) • λ2 λ4 λ3 • (a3 , T3 , I3 , F3 ) (a4 , T4 , I4 , F4 ) • Fig. 10. Derived SVN-G C5 .

0 Tjjþ1 þ Ijjþ1 þ Fjjþ1 3. Let C5 be a derivable single-valued neutrosophic graph and H ¼ ðV ; fEj gj¼1 Þ be an associated single-valued neutrosophic hypergraph with single-valued neutrosophic graph C . Since ða Þ ða Þ ¼ ðac Þ; ða Þ and 5

1

i

2

1

2

ða1 Þ i ða5 Þ ¼ ðac 1 Þ; ða5 Þ, we get k 2 N, E1 ; E2 ; E3 H so that a1 2 E1 ; jE1 j ¼ k; a2 2 E2 ; jE2 j ¼ k þ i; a5 2 E3 and jE3 j ¼ k þ i. On the other hand, ða Þ ða Þ ¼ ðac ;, Þ; ða Þ, ða Þ ða Þ ¼ ðac Þ; ða Þ a n d ða Þ ða Þ ¼ b 3

i

2

3

2

3

i

4

3

3

4

i

1

implies that there exist E4 ; E5 H so that a3 2 E4 ; jE4 j ¼ jE1 j, a4 2 E5 and jE2 j ¼ jE3 j ¼ jE5 j ¼ k þ i. Since jj ða4 Þj j ða5 Þjj 6¼ i, jj ða3 Þj j ða5 Þjj ¼ i and jj ða4 Þj j ða1 Þjj ¼ i, we get that ða4 Þ i ða5 Þ ¼ b ;, ða3 Þ i ða5 Þ ¼ ðac 3 Þ; ða5 Þ and ða Þ ða Þ ¼ ðac Þ; ða Þ. But H ¼ ðV ; EÞ, where E ¼ fE ; E ; E ; E ; E g, H= 4

i

1

4

1

1

2

3

4

5

ffi C5 and jE= j ¼ 6, which is a contradiction. Therefore, C5 cannot be a derivable single-valued neutrosophic graph.

Proposition 4.10. Let 6 n 2 N. Then ðCn ; A; BÞ is not a derivable single-valued neutrosophic graph.

Proof. Since for any 6 n 2 N, Cn is homeomorphic to ðC5 ; A; BÞ, by Lemma 4.9, ðCn ; A; BÞ is not a derivable single-valued neutrosophic graph.

Theorem 4.11. Let Cn ¼ ðV ; E; A; BÞ be a cycle single-valued neutrosophic graph.

Then Cn is a derivable single-valued neutrosophic graph if and only if n ¼ 3 and n ¼ 4.

Proof. If n ¼ 3, then C3 ffi K3 and so by Theorem 4.7, C3 is a derivable singlevalued neutrosophic graph. If n ¼ 4, then by Example 4.8, C4 is a derivable singlevalued neutrosophic graph. By Proposition 4.10, the converse, is obtained.

Corollary 4.12. Let G ¼ ðV ; E; A; BÞ be a single-valued neutrosophic non-complete graph in which has cycle. Then G ¼ ðV ; E; A; BÞ is a non-derivable single-valued neutrosophic graph if and only if ðiÞ G 6ffi C3 and G 6ffi C4 ; ðiiÞ it contains a single-valued neutrosophic subgraph that is homeomorphic to Cn , where 3; 4 6¼ n.

Achievable Single-Valued Neutrosophic Graphs in Wireless Sensor Networks

169

(a1 , 0.1, 0.2, 0.3) • λ6 λ1 • (a2 , 0.2, 0.3, 0.4) (a6 , 0.6, 0.7, 0.8) • λ5 λ2 • (a3 , 0.3, 0.4, 0.5) (a5 , 0.5, 0.6, 0.7) • λ4

λ3 • (a4 , 0.4, 0.5, 0.6) Fig. 11. Derived cycle SVN-G C6 .

(a1 , 0.1, 0.2, 0.3) • λ6 λ1 λ7 • (a2 , 0.2, 0.3, 0.4) (a6 , 0.6, 0.7, 0.8) • λ5 λ8 λ9 λ2 • (a3 , 0.3, 0.4, 0.5) (a5 , 0.5, 0.6, 0.7) • λ4

λ3 • (a4 , 0.4, 0.5, 0.6) Fig. 12. SVN-G C 6% .

Fig. 13. SVN-HG.

,0 .5 ,0 .6 ,0 .7 ) (a 5

(b1 , 0.2, 0.3, 0.4)

(a2 , 0.8, 0.1, 0.2)

(b2 , 0.4, 0.5, 0.6)

(a4 , 0.5, 0.5, 0.5)

(a1 , 0.1, 0.2, 0.3)

(a3 , 0.3, 0.4, 0.5)

(a6 , 0.7, 0.3, 0.8) (b3 , 0.6, 0.7, 0.6)

Example 4.13. Consider the cycle single-valued neutrosophic graph C6 in Fig. 11, where 1 ¼ ð0:1; 0:3; 0:4Þ; 2 ¼ ð0:2; 0:4; 0:5Þ; 3 ¼ ð0:3; 0:5; 0:6Þ; 4 ¼ ð0:4; 0:6; 0:7Þ; 5 ¼ ð0:5; 0:7; 0:8Þ and 6 ¼ ð0:1; 0:7; 0:8Þ: By Proposition 4.10, C6 is not a derivable single-valued neutrosophic graph. Now we add some edges to C6 as Fig. 12, where 7 ¼ ð0:1; 0:5; 0:6Þ; 8 ¼ ð0:2; 0:6; 0:7Þ a n d 9 ¼ ð0:3; 0:7; 0:8Þ: H en ce n ow w e consider the single-valued neutrosophic hypergraph G ¼ ðV ; fEj g nj¼1 Þ in Fig. 13.

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Clearly ða1 Þ ¼ fa1 g; ða2 Þ ¼ fa2 ; b1 g; ða3 Þ ¼ fa3 g; ða4 Þ ¼ fa4 ; b2 g; ða5 Þ ¼ fa5 g; ða6 Þ ¼ fa6 ; b3 g and it is easy to compute that H= ffi C 6% .

In Example 4.13, we saw that C6 is not a derivable single-valued neutrosophic graph, while we added some edges to this single-valued neutrosophic graph and converted to a derivable single-valued neutrosophic graph. Duo to this problem we will have the following de¯nition.

De¯nition 4.14. Let G ¼ ðV ; E; A; BÞ be a non derivable single-valued neutrosophic graph and i 6¼ 0. We will call non-single-valued neutrosophic complete graph G % ¼ ðV ; E % ; A % ; B % Þ is an extended derivable single-valued neutrosophic graph of G, if E % is obtained by adding the least number of edges to E such that G % be a derivable single-valued neutrosophic graph. Also we will say G ¼ ðV ; E; A; BÞ is an extended derivable single-valued neutrosophic graph.

Example 4.15. By Example 4.13, cycle single-valued neutrosophic graph C6 is an extended derivable single-valued neutrosophic graph.

Theorem 4.16. ðC5 ; A; BÞ is not an extended derivable single-valued neutrosophic graph.

Proof. Since i 1, by Lemma 4.9, for any single-valued neutrosophic hypergraph H ¼ ðV ; EÞ, where E ¼ fE1 ; E2 ; E3 ; E4 ; E5 g, we get that H= has a cycle of maximum length 4. By adding any edges to C5 , since jV j is odd again, duo to Lemma 4.9, we get that jE= j ¼ 6, else all edges in H= be connected that implies i ¼ 0, which is a contradiction.

Corollary 4.17. Let k 2 N. Then C2kþ1 is not an extended derivable single-valued neutrosophic graph.

Theorem 4.18. Let k 2 N. Then ðiÞ C2k ¼ ðV ; E % ; A % ; B % Þ is an extended derivable single-valued neutrosophic graph; ðiiÞ jE % j ¼ k 2 .

Proof. ði; iiÞ Let V ¼ V1 [ V2 , where V1 ¼ fa1 ; a3 ; a5 ; . . . ; a2k 1 g, V2 ¼ fa2 ; a4 ; a6 ; . . . ; a2k g and for any 1 j 2k; ej ¼ ac j ; ajþ1 . Now for any j 2 f1; 3; 5; . . . ; a2k 1 g consider Ej so that aj 2 Ej , jEj j ¼ kj , ajþ1 2 Ejþ1 and jEjþi j ¼ kj þ i, where kj 2 N. A simple computation shows that H ¼ ðV ; fEs ¼ fðvj ; TEs ðvj Þ; IEs ðvj Þ; FEs ðvj ÞÞggs¼1 Þ is a single-valued neutrosophic hypergraph, where V ¼ V [ W and W is any set so that jW j ¼ iðn=2Þ. Moreover, by de¯nition of , we can see that ðaj Þ ¼ Ej

d and ðaj Þ i ðajþ1 Þ ¼ ða j Þ; ðajþ1 Þ, whence 1 j 2k. Since jV1 j ¼ jV2 j ¼ n=2, d d Þ; ða Þ; ða Þ ða Þ ¼ ða Þ; ða Þ, for any j 6¼ 1; ða Þ ða Þ ¼ ða j

i

jþ1

ða1 Þ i ða2 Þ ¼ d ða1 Þ; ða2 Þ jE % j ¼ ð2k=2Þðð2kÞ=2 2Þ þ 2k

j

ða

and ¼ k 2.

jþ1

1 Þ i ða2k Þ

j

¼

i

j 1

d ða1 Þ; ða2k Þ,

j

j 1

we get that

Achievable Single-Valued Neutrosophic Graphs in Wireless Sensor Networks

171

4.2. Derivable single-valued neutrosophic trees In this section, we study derivable regular single-valued neutrosophic trees and introduce Y -single-valued neutrosophic tree T6 which is not a derivable single-valued neutrosophic tree.

Lemma 4.19. Y -single-valued neutrosophic tree T6 is not a derivable single-valued neutrosophic graph.

Proof. Let V ¼ fa1 ; a2 ; a3 ; a4 ; a5 ; a6 g. Consider regular single-valued neutrosophic tree T ¼ ðV ; E; A; BÞ in Fig. 14, where for any 1 j 6 we have 0 Tj þ Ij þ Fj 3, j ¼ ðTjjþ1 ; Ijjþ1 ; Fjjþ1 Þ and 0 Tjjþ1 þ Ijjþ1 þ Fjjþ1 3.

Let Y -single-valued neutrosophic tree T6 (Fig. 14), be a derivable regular singlevalued neutrosophic graph and H ¼ ðV , fEs ¼ fðvj ; TEs ðvj Þ, IEs ðvj Þ, FEs ðvj ÞÞggs¼1 Þ be an associated single-valued neutrosophic hypergraph with regular singlevalued neutrosophic graph Y -single-valued neutrosophic tree T6 . Since for any 1 j 4, ða Þ ða Þ ¼ ðac Þ; ða Þ, ða Þ ða Þ ¼ ðac Þ; ða Þ and ða Þ ða Þ ¼ 2

i

j

2

3

j

i

5

3

5

4

i

6

ðac 4 Þ; ða6 Þ, we get k 2 N, E1 ; E2 ; E3 ; E4 H so that a1 2 E1 ; jE1 j ¼ k; a2 2 E2 ; jE2 j ¼ k þ i; a3 ; a4 2 E3 , jE3 j ¼ k þ 2i a5 ; a6 2 E4 and jE4 j ¼ k þ 3i. It follows that, ða Þ ða Þ ¼ ðac Þ; ða Þ and ða Þ ða Þ ¼ ðac Þ; ða Þ, which is a contradiction.

3

i

6

3

4

6

i

5

4

5

Therefore Y -single-valued neutrosophic tree T5 cannot be a derivable single-valued neutrosophic graph.

Theorem 4.20. Let T ¼ ðV ; E; A; BÞ be a regular single-valued neutrosophic tree and for any v 2 V ; degðvÞ 2. Then T is a derivable single-valued neutrosophic graph.

λ4

λ3

(a 3

•

• (a

(a6 , T6 , I6 , F6 )

4,

T4

,T

3,

,I

I3

4,

F

,F

4)

3)

Proof. Let i 2 N; G ¼ ðV ; E; A; B; 0 Þ be a regular single-valued neutrosophic graph with no cycle, in such a way V ¼ fa1 ; a2 ; . . . ; an g and E ¼ fe1 ; e2 ; . . . ; em g, where m n. Suppose that for any 1 j 6¼ j 0 m; ejj 0 ¼ faj ; aj 0 g ¼ aj 0 aj 0 . De¯ne a

λ2

•

λ5

•

(a5 , T5 , I5 , F5 )

•(a2 , T2 , I2 , F2 ) λ1 (a1 , T1•, I1 , F1 ) Fig. 14. Y-SVN-T T6 .

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M. Hamidi & A. Borumand Saeid

single-valued neutrosophic hypergraph G ¼ ðV ; fEs g ns¼1 Þ as follows: Es ¼ fðaj ; TEs ðaj Þ; IEs ðaj Þ; FEs ðaj ÞÞg [ Cj ¼ fða 0j ; TCs ða 0j Þ; ICs ða 0j Þ; FCs ða 0j ÞÞg such that jC1 j ¼ 1, for any 1 < l n, jClþ1 j jCl j ¼ i, for any 1 j; j 0 n we have Cj \ Cjþ1 ¼ ; and TA ðvÞ ¼ ^ðTEs ðvÞ ^ TCs ðvÞÞ; IA ðvÞ ¼ _ðIEs ðvÞ _ ICs ðvÞÞ and FA ðvÞ ¼ _ðFEs ðvÞ _ FCs ðvÞÞ: It is easy to see that for any 1 < j; j 0 n, jEjþ1 j jEj j ¼ i and S Ej \ Ej 0 ¼ ;. A simple computation shows that V ¼ nj¼1 ðCj [ V Þ and ðV ; fEj g nj¼1 Þ is a single-valued neutrosophic hypergraph. Clearly for any 1 j n; ððaj ; TEs ðaj Þ; IEs ðaj Þ; FEs ðaj ÞÞÞ ¼ Ej and since Ej \ Ej 0 ¼ ;, we get that G= ¼ f ðaj Þ ¼ ððaj ; TEs ðaj Þ; IEs ðaj Þ; FEs ðaj ÞÞÞ j 1 j ng and so for any i 2 N obtain ( ðac r Þ; ðas Þ if jr sj ¼ i; ðar Þ i ðas Þ ¼ b ; if jr sj 6¼ i: de¯ne map ’ : ðG= ; iÞ ! G ¼ ðV ; EÞ by ’ð ðaj ÞÞ ¼ aj and c ’ðð ðaj Þ; ðaj 0 ÞÞÞ ¼ ejj 0 . Let aj ; aj 0 2 V . If ðaj Þ ¼ ðaj 0 Þ, then jEj j ¼ jEj 0 j and so

Now,

Ej ¼ Ej 0 . Thus, ’ð ðaj ÞÞ ¼ ’ð ðaj 0 ÞÞ. Since for any 1 j 6¼ j 0 n; 0 0 ’ð ðaj Þ i ðaj 0 ÞÞ ¼ ’ðð ðac j Þ; ðaj 0 ÞÞÞ ¼ ejj 0 ¼ aj aj 0 ¼ ’ð ðaj ÞÞ ’ð ðaj 0 ÞÞ;

in other words, if ðaj Þ and ðaj 0 Þ in G= are adjacent, then ’ð ðaj ÞÞ and ’ð ðaj 0 ÞÞ in G are adjacent. So ’ is a homomorphism. It is easy to see that ’ is bijection and so is an isomorphism. It follows that any single-valued neutrosophic graph is a derivable single-valued neutrosophic graph.

Corollary 4.21. Let n 5. Then Tn is a derivable single-valued neutrosophic graph.

Example 4.22. Let V ¼ fa1 ; a2 ; a3 ; a4 g, E ¼ fe1 ; e2 ; e3 g. Consider single-valued neutrosophic tree T4 in Fig. 15: S 0 0 For any arbitrary set B ¼ 10 i¼1 bi where for any j; j have aj 6¼ b j , de¯ne a singlevalued neutrosophic hypergraph G ¼ ðV ; fEj gÞ in a way that E1 ¼ fða1 ; 0:9; 0:1; 0:2Þg [ fðb1 ; 0:6; 0:2; 0:4Þg; E2 ¼ fða2 ; 0:8; 0:1; 0:1Þg [ fðb2 ; 0:3; 0:1; 0:1Þ; ðb3 ; 0:9; 0:2; 0:1Þg; E3 ¼ fða3 ; 0:7; 0:2; 0:1Þg [ fðb4 ; 0:8; 0:1; 0:1Þ; ðb5 ; 0:7; 0:1; 0:3Þ; ðb6 ; 0:9; 0:4; 0:2Þg (a1 , 0.6, 0.2, 0.4) • (a3 , 0.7, 0.4, 0.3) •

(0.3, 0.2, 0.4) , 0.3) (0.3, 0.4 (0.5, 0.4, 0.4) Fig. 15. SVN-T T4 .

• (a2 , 0.3, 0.2, 0.1) • (a4 , 0.5, 0.1, 0.4)

173

(a2 , 0.8, 0.1, 0.1) (b2 , 0.3, 0.1, 0.1) (b3 , 0.9, 0.2, 0.1)

(a3 , 0.7, 0.2, 0.1) (b4 , 0.8, 0.1, 0.1) (b5 , 0.7, 0.1, 0.3) (b6 , 0.9, 0.4, 0.2)

(a4 , 0.5, 0.1, 0.2) (b7 , 0.8, 0.1, 0.3) (b8 , 0.9, 0.1, 0.1) (b9 , 1, 0.1, 0.2) (b10 , 0.9, 0.1, 0.4)

Achievable Single-Valued Neutrosophic Graphs in Wireless Sensor Networks

(a1 , 0.9, 0.1, 0.2) (b1 , 0.6, 0.2, 0.4) Fig. 16. SVN-HG.

and E4 ¼ fða4 ; 0:5; 0:1; 0:2Þg [ fðb7 ; 0:8; 0:1; 0:3Þ; ðb8 ; 0:9; 0:1; 0:1Þ; ðb9 ; 1; 0:1; 0:2Þ; ðb10 ; 0:9; 0:1; 0:4Þg: Hence consider the single-valued neutrosophic hypergraph G ¼ ðV ; fEi g ni¼1 Þ in Fig. 16. By Theorem 4.20, for i ¼ 1 and any 1 j 4; ðaj Þ ¼ Ej , so G= ¼ ðf ða1 Þ; ða2 Þ; ða3 Þ; ða4 Þg; ff ða1 Þ; ða2 Þg; f ða2 Þ; ða3 Þg; f ða3 Þ; ða4 ÞggÞ is a single-valued neutrosophic tree in Fig. 17. It is easy to see that T4 ffi G= and so T4 ¼ ðV ; E; A; BÞ is a derivable single-valued neutrosophic graph. Moreover, for any 2 i 2 N, we can construct another associated single-valued neutrosophic hypergraphs of single-valued neutrosophic graph G. Theorem 4.23. Let Tn ¼ ðV ; E; A; BÞ be a regular single-valued neutrosophic tree which is not contain a single-valued neutrosophic subtree, that is homeomorphic to Y -single-valued neutrosophic tree T6 and n 6. Then Tn is a derivable single-valued neutrosophic tree.

Proof. Let V ¼ fa1 ; a2 ; . . . ; an g. We rearrange the regular single-valued neutrosophic tree T by n 1 dn . . . d3 d2 d1 , where di ¼ degðai Þ. Since degða1 Þ ¼ d1 , we get that aj1 ; aj2 ; . . . ; ajd1 1 ; E1 ; E2 so that a1 2 E1 ; aj1 ; aj2 ; . . . ; ajd1 1 2 E2 and (ρ(a1 ), 0.6, 0.2, 0.4) • (ρ(a3 ), 0.7, 0.4, 0.3) •

(0.3, 0.2, 0.4) , 0.3) (0.3, 0.4 (0.5, 0.4, 0.4) Fig. 17. Derived SVN-G for i ¼ 1.

• (ρ(a2 ), 0.3, 0.2, 0.1) • (ρ(a4 ), 0.5, 0.1, 0.4)

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jjE1 j jE2 jj ¼ i. By Lemma 4.19, for any 1 j 0 d1 1, degðajj 0 Þ ¼ 1 and by rearrangement ajd1 ¼ a2 . If degðajd1 Þ ¼ 1 the proof is obtained. If degðajd1 Þ > 1 in a similar way, there exist ar1 ; ar2 ; . . . ; ajd2 1 ; E3 ; E4 so that a2 2 E3 ; ar1 ; ar2 ; . . . ; ard1 1 2 E4 and jjE3 j jE4 jj ¼ i. By Lemma 4.19, for any 1 r 0 d2 1, degðarr 0 Þ ¼ 1 and by rearrangement ard2 ¼ a3 . If degðard2 Þ ¼ 1 the proof is obtained. If degðard2 Þ > 1 we can continue. Since jV j < 1, then this process stops. A simple computation shows that H ¼ ðV [ V 0 ; fEs ¼ fðaj ; TEs ðaj Þ; IEs ðaj Þ; FEs ðaj ÞÞggs¼1 Þ is a single-valued neutrosophic hypergraph, where for any 1 j n, ð ðaj Þ; T ðaj Þ ; I ðaj Þ ; F ðaj Þ Þ and ðaj ; Taj ; Iaj ; Faj Þ T ðaj Þ ¼ Taj ; I ðaj Þ ¼ Iaj and F ðaj Þ ¼ Faj .

By Theorem 4.20, H= ffi ðTn ; A; BÞ.

Corollary 4.24. Let T ¼ ðV ; E; A; BÞ be a regular single-valued neutrosophic tree.

Then T ¼ ðV ; E; A; BÞ is a non-derivable single-valued neutrosophic graph if and only if it contains a single-valued neutrosophic subtree that is homeomorphic to Y -singlevalued neutrosophic tree ðT6 ; A; BÞ.

Example 4.25. Consider the single-valued neutrosophic tree T 5 ¼ ðV ; E; A; BÞ in Fig. 18. Now construct the single-valued neutrosophic hypergraph H in Fig. 19. By Theorem 4.20, for i ¼ 1 , we have ða; 0:4; 0:5; 0:6Þ ¼ fða; 0:4; 0:5; 0:6Þg; ðb; 0:1; 0:2; 0:3Þ ¼ fðb; 0:1; 0:2; 0:3Þ; ðb1 ; 0:1; 0:2; 0:3Þg; ðc; 0:7; 0:8; 0:9Þ ¼ fðc; 0:7; 0:8; 0:9Þ; ðb2 ; 0:7; 0:8; 0:9Þg; ðd; 0:2; 0:4; 0:6Þ ¼ fðd; 0:2; 0:1; 0:3Þ; ðb3 ; 0:2; 0:1; 0:3Þg and ðe; 0:3; 0:5; 0:7Þ ¼ fðe; 0:4; 0:5; 0:6Þ; ðb4 ; 0:4; 0:5; 0:6Þg: So H= ffi ðT5 ; V ; E; A; BÞ and so ðT5 ; V ; E; A; BÞ is a derivable single-valued neutrosophic tree.

Theorem 4.26. Let T ¼ ðV ; E; A; BÞ be a single-valued neutrosophic tree and i 2 N . If H ¼ ðV ; fEj gj Þ is a single-valued neutrosophic hypergraph of single-valued neutrosophic tree T , then ðV = Þ ¼ jEj i.

(0.3, 0.5, 0.7)

6) 5, 0. .

0 , (0.1

• (b, 0.1, 0.2, 0.3)

• (a, 0.4, 0.5, 0.6) (0.2, 0.5, 0.6)

(e, 0.3, 0.5, 0.7)

•

(0 .4

,0

.8,

• (d, 0.2, 0.4, 0.6) Fig. 18. SVN-T T5 .

0.9

)

•

(c, 0.7, 0.8, 0.9)

175

(e, 0.4, 0.5, 0.6) (b4 , 0.3, 0.5, 0.7)

(d, 0.2, 0.1, 0.3) (b3 , 0.5, 0.4, 0.6)

(b2 , 0.7, 0.8, 0.5)

(c, 0.9, 0.3, 0.9)

(b, 0.7, 0.1, 0.2) (b1 , 0.1, 0.2, 0.3)

(a, 0.4, 0.5, 0.6)

Achievable Single-Valued Neutrosophic Graphs in Wireless Sensor Networks

Fig. 19. SVN-HG.

Proof. Let V ¼ fa1 ; a2 ; . . . ; an g, for any 1 j n; degðaj Þ ¼ dj and ðGÞ ¼ dp , where 1 p n: Set E1 ¼ fap g; E2 ¼ faj j j 6¼ p and ajd ; ap 6¼ b ;g so that jE2 j ¼ i þ 1; Ek ¼ fak j ak 62 E1 [ E2 g and jEk j ¼ 1. A simple calculation shows that H ¼ ðV ; fEi gi Þ is a single-valued neutrosophic hypergraph. Since jV j ¼ n, we get jfEj ; jEj j ¼ 1gj ¼ n ði þ 1Þ and so single-valued neutrosophic hypergraph H ¼ ðV ; fEi gi Þ has ðn iÞ hyperedges. It is easy to see that V = ¼ f ðap Þ; ðak Þ;

ðak 0 Þ j ak 2 Ek ; ak 0 62 E1 [ E2 g;

j ðap Þj ¼ 1, for any ak 2 Ek we have j ðak Þj ¼ i þ 1 and for any ak 0 62 E1 [ E2 ; j ðak 0 Þj ¼ 1. So for any r ¼ 6 r 0, ( b ; if far ; ar 0 g E1 [ Ek [ Ek 0 ; ðar Þ i ðar 0 Þ ¼ ðarc Þ; ðar 0 Þ otherwise: Since jV j ¼ n and T is a single-valued neutrosophic tree, we get that jEj ¼ n 1 and so ðV = Þ ¼ n i 1 ¼ jEj i.

Example 4.27. Let V ¼ f1; 2; 3; 4; 5; 6; 7g. Consider single-valued neutrosophic tree T ¼ ðV ; E; A; BÞ in Fig. 20. Since ðT Þ ¼ 4, we obtain i ¼ 3 and the single-valued neutrosophic hypergraph in Fig. 21. Hence, V = is a single-valued neutrosophic tree with 4 vertices and 3 edges such that ðV = Þ ¼ 3 (Fig. 22).

Corollary 4.28. Let n 2 N. (i) If n 2, then any single-valued neutrosophic tree Tn is not a self derivable singlevalued neutrosophic graph; (ii) Cn is not a self derivable single-valued neutrosophic graph.

Proof. ðiÞ Let jV j ¼ n, Tn ¼ ðV ; EÞ and Tn be a self derivable single-valued neutrosophic graph. By Theorem 4.26, Tn is a self derivable single-valued neutrosophic graph if and only if jEj i ¼ jEj if and only if i ¼ 0. By Theorem 4.7, must Tn ffi Kn where is a contradiction. ðiiÞ Since Cn has cycle, by Theorem 4.7, is clear.

M. Hamidi & A. Borumand Saeid

(2, 0.2, 0.3, 0.4) • , .2 (0

(6, 0.6, 0.7, 0.8) •

0.6,

(0 .5 ,

(0.3, 0.5, 0.6)

(0.4 ,

0.

9)

(0.5, 0 .7

6) 0.

, 0.8)

5, 0.

(4, 0.4, 0.5, 0.6) •

(1, 0.1, 0.2, 0.3) • (0.1, 0.5, 0.6)

(7, 0.7, 0.8, 0.9) •

0.

8,

176

0.7)

• (5, 0.5, 0.6, 0.7)

• (3, 0.3, 0.4, 0.5)

(7, 0.7, 0.8, 0.9)

(6, 0.6, 0.7, 0.8)

(4, 0.4, 0.5, 0.6)

(1, 0.6, 0.2, 0.2) (2, 0.5, 0.1, 0.3) (3, 0.2, 0.1, 0.1) (5, 0.1, 0.1, 0.2)

Fig. 20. SVN-T T .

Fig. 21. SVN-HG.

η ∗ (1) •

• η ∗ (7)

• η ∗ (6)

• η ∗ (4)

Fig. 22. Derived SVN-G G= for i ¼ 3.

4.3. Derivable single-valued neutrosophic complete graphs In this section, we consider regular single-valued neutrosophic complete graphs, regular single-valued neutrosophic complete bigraphs and we investigate their associated single-valued neutrosophic hypergraphs.

Theorem 4.29. Let G ¼ ðV ; E; A; BÞ be a derivable single-valued neutrosophic complete graph via associated single-valued neutrosophic hypergraph H and jV j ¼ n.

Achievable Single-Valued Neutrosophic Graphs in Wireless Sensor Networks

177

Then ðiÞ if H is a discrete complete single-valued neutrosophic hypergraph, where their hyperedges are 2 k-hyperedge, then i ¼ 0 ; ðiiÞ if H is a discrete complete single-valued neutrosophic hypergraph, where their hyperedges are 2 k-hyperedge, then jHj ¼ kn.

Proof. Clearly H= ffi G. ðiÞ Since H= is a single-valued neutrosophic complete graph, we get that for any ðx; TEi ðxÞ; IEi ðxÞ; FEi ðxÞÞ; ðy; TEi ðyÞ; IEi ðyÞ; FEi ðyÞÞ 2 H= ; ðx; TEi ðxÞ; IEi ðxÞ; FEi ðxÞÞ i ðy; TEi ðyÞ; IEi ðyÞ; FEi ðyÞÞ c ðy; TEi ðyÞ; IEi ðyÞ; FEi ðyÞÞ ¼ ðx; TEi ðxÞ; IEi ðxÞ; FEi ðxÞÞ; and so jj ðx; TEi ðxÞ; IEi ðxÞ; FEi ðxÞÞj j ðy; TEi ðyÞ; IEi ðyÞ; FEi ðyÞÞjj ¼ i. On the other hand, H is a discrete complete single-valued neutrosophic hypergraph, where 2 k-hyperedge, then for any ðx; TEi ðxÞ; IEi ðxÞ; FEi ðxÞÞ, ðy; TEi ðyÞ; IEi ðyÞ, FEi ðyÞÞ 2 H= , j ðx; TEi ðxÞ; IEi ðxÞ; FEi ðxÞÞj ¼ j ðy; TEi ðyÞ, IEi ðyÞ; FEi ðyÞÞj ¼ k. It follows that i ¼ jj ðx; TEi ðxÞ; IEi ðxÞ; FEi ðxÞÞj j ðy; TEi ðyÞ; IEi ðyÞ; FEi ðyÞÞjj ¼ jk kj ¼ 0. ðiiÞ Since for any ðx; TEi ðxÞ; IEi ðxÞ; FEi ðxÞÞ; ðy; TEi ðyÞ; IEi ðyÞ; FEi ðyÞÞ 2 H= ; j ðx; TEi ðxÞ; IEi ðxÞ; FEi ðxÞÞj ¼ j ðy; TEi ðyÞ; IEi ðyÞ; FEi ðyÞÞj ¼ k and n ¼ jH= j ¼ jHj=j j, we get that jHj ¼ nj j ¼ nk.

Theorem 4.30. Let H ¼ ðV ; fEi ¼ fðaj ; TEi ðaj Þ; IEi ðaj Þ; FEi ðaj ÞÞgg m i¼1 Þ be a discrete complete single-valued neutrosophic hypergraph, where 2 r-hyperedge and jHj ¼ rn. Then ðiÞ if i ¼ 0 , then its derivable single-valued neutrosophic graph is isomorphic to regular single-valued neutrosophic complete graph Kn=r ; ðiiÞ if for n 2 N i ¼ n , then its derivable single-valued neutrosophic graph is isomorphic to single-valued neutrosophic complete graph K n=r .

Proof. Since H is a discrete complete single-valued neutrosophic hypergraph, where 2 r-hyperedge, then for all ðxÞ ¼ ðx; TEi ðxÞ; IEi ðxÞ; FEi ðxÞÞ; ðyÞ ¼ ðy; TEi ðyÞ; IEi ðyÞ; FEi ðyÞÞ 2 H= ; j ðxÞj ¼ j ðyÞj ¼ r. c ðiÞ If i ¼ 0 , then for any ðxÞ; ðyÞ 2 H= ; ðxÞ 0 ðyÞ ¼ ðxÞ; ðyÞ. Since jHj ¼ n we get that jH= j ¼ n=r. ðiiÞ If for any n 2 N; i ¼ n , then for any ðxÞ; ðyÞ 2 H= ; ðxÞ n ðyÞ ¼ b ;. Hence H= is null single-valued neutrosophic graph and jHj ¼ n implies that jH= j ¼ n=r.

Theorem 4.31. Let G ¼ ðV ; E; A; BÞ be a derivable regular single-valued neutrosophic graph by i ¼ 0 and jV j ¼ n. Then ðiÞ if G ¼ ðV ; E; A; BÞ is a connected single-valued neutrosophic graph, then there exists n 2 N so that G ffi Kn ,

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ðiiÞ if G ¼ ðV ; E; A; BÞ is a non-connected single-valued neutrosophic graph, then all connected components of G as G 0 are isomorphic to Kk or K k whence, 1 k n 2 N .

Proof. Since G ¼ ðV ; E; A; BÞ is a derivable regular single-valued neutrosophic graph, then we have a single-valued neutrosophic hypergraph H ¼ ðV ; fEi ¼ fðaj ; TEi ðaj Þ; IEi ðaj Þ; FEi ðaj ÞÞgg m i¼1 Þ as associated single-valued neutrosophic hypergraph to G, where m n, T ðEi Þ ð ðxÞÞ ¼ TA ðxÞ, I ðEi Þ ð ðxÞÞ ¼ IA ðxÞ and F ðEi Þ ð ðxÞÞ ¼ FA ðxÞ. ðiÞ Since G ¼ ðV ; E; A; BÞ is a connected derivable regular single-valued neutrosophic graph, jV j ¼ n and i ¼ 0 , we get that there exist a1 ; a2 ; . . . ; an 2 H so that for any 1 j; j 0 n; ða Þ ða 0 Þ ¼ ðac Þ; ða 0 Þ. j

i

j

j

j

ðiiÞ Since G ¼ ðV ; E; A; BÞ is a non-connected derivable single-valued neutrosophic graph, then there exist a1 ; . . . ; an 2 H such that for any 1 j; j 0 n; j ðaj Þj 6¼ j ðaj 0 Þj or j ðaj Þj ¼ j ðaj 0 Þj. Let 1 l m and P l ¼ f ðaj Þ j j ðaj Þj ¼ lg, P 0 then for any 1 l m, it is easy to see that m l¼1 ðl:jP l jÞ ¼ m. If G be a connected 0 components of G, then there exists 1 l m so that G ffi KjP l j .

Theorem 4.32. Let m; n 2 N. Then for any 1 i there exists a partitioned singlevalued neutrosophic hypergraph H such that H= ffi Km;n , where Km;n is a regular single-valued neutrosophic complete bigraph.

Proof. Let 1 i. Then we consider t 2 N and a partitioned single-valued neutrosophic hypergraph H ¼ ðV ; fEj gj Þ, where has hyperedges E1 ; E2 ; . . . ; Em that for any 1 j m; jEj j ¼ t and hyperedges Emþ1 ; Emþ2 ; . . . ; Enþm that for any m þ 1 j n; jEj j ¼ t þ i. Let for any 1 j m; Ej ¼ fða 1j ; TEi ða 1j Þ; IEi ða 1j Þ; FEi ða 1j ÞÞ; ða 2j ; TEi ða 2j Þ; IEi ða 2j Þ; FEi ða 1j ÞÞ; . . . ; ða tj ; TEi ða tj Þ; IEi ða tj Þ; FEi ða tj ÞÞg and for any m þ 1 j n þ m; Ej ¼ fða 1j ; TEi ða 1j Þ; IEi ða 1j Þ; FEi ða 1j ÞÞ; ða 2j ; TEi ða 2j Þ; IEi ða 2j Þ; FEi ða 2j ÞÞ; tþi tþi . . . ; ða 1j ; TEi ða tþi j Þ; IEi ða j Þ; FEi ða j ÞÞg:

A simple calculation shows that for any 1 j m, for any 1 j 0 t; 0

0

0

0

ðða jj ; TEi ða jj Þ; IEi ða jj Þ; FEi ða jj ÞÞÞ ¼ fða 1j ; TEi ða 1j Þ; IEi ða 1j Þ; FEi ða 1j ÞÞ; ða 2j ; TEi ða 2j Þ; IEi ða 2j Þ; FEi ða 2j ÞÞ; . . . ; ða tj ; TEi ða tj Þ; IEi ða tj Þ; FEi ða tj ÞÞg and for any m þ 1 l n þ m, for any 1 l 0 t þ i; 0

0

0

0

ðða ll ; TEi ða ll Þ; IEi ða ll Þ; FEi ða ll ÞÞÞ ¼ fða 1l ; TEi ða 1l Þ; IEi ða 1l Þ; FEi ða 1l ÞÞ; tþi tþi tþi ða 2l ; TEi ða 2l Þ; IEi ða 2l Þ; FEi ða 2l ÞÞ; . . . ; ða tþi l ; TEi ða l Þ; IEi ða l Þ; FEi ða l ÞÞg:

Achievable Single-Valued Neutrosophic Graphs in Wireless Sensor Networks

179

So H= ¼ f ðða 1r ; TEi ða 1r Þ; IEi ða 1r Þ; FEi ða 1r ÞÞÞ; ðða 1s ; TEi ða 1s Þ; IEi ða 1s Þ; FEi ða 1s ÞÞÞ j 1 r m

and

m þ 1 s n þ mg;

whence for any 1 r m; j ðða 1r ; TEi ða 1r Þ; IEi ða 1r Þ; FEi ða 1r ÞÞÞj ¼ t and for any m þ 1 s n þ m; j ðða 1s ; TEi ða 1s Þ; IEi ða 1s Þ; FEi ða 1s ÞÞÞj ¼ t þ i. Thus for any 1 r m; and for any m þ 1 s n þ m we have ðða 1r ; TEi ða 1r Þ; IEi ða 1r Þ; FEi ða 1r ÞÞÞ i ðða 1s ; TEi ða 1s Þ; IEi ða 1s Þ; FEi ða 1s ÞÞÞ c ðða 1s ; TEi ða 1s Þ; IEi ða 1s Þ; FEi ða 1s ÞÞÞ; ¼ ðða 1r ; TEi ða 1r Þ; IEi ða 1r Þ; FEi ða 1r ÞÞÞ; for any 1 r m; ðða 1r ; TEi ða 1r Þ; IEi ða 1r Þ; FEi ða 1r ÞÞÞ i ðða 1s ; TEi ða 1s Þ; IEi ða 1s Þ; FEi ða 1s ÞÞÞ ¼ b ; and for any m þ 1 s n þ m; ;: ðða 1s ; TEi ða 1s Þ; IEi ða 1s Þ; FEi ða 1s ÞÞÞ i ðða 1s ; TEi ða 1s Þ; IEi ða 1s Þ; FEi ða 1s ÞÞÞ ¼ b Therefore H= ffi ðKm;n ; A; BÞ.

5. Wireless Sensor (Hyper) Networks and Achievable Single-Valued Neutrosophic (Hyper) Graphs In this section, we describe some applications of achievable single-valued neutrosophic graphs.

The study of complex networks plays a main role in the important area of multidisciplinary research involving physics, chemistry, biology, social sciences, and information sciences. These systems are commonly represented by means of simple or directed graphs that consist of sets of nodes representing the objects under investigation, e.g., people or groups of people, molecular entities, computers, etc., joined together in pairs by links if the corresponding nodes are related by some kind of relationship. These networks include the internet, the world wide web, social networks, information networks, neural networks, food webs, and protein{protein interaction networks. In some cases, the use of simple or directed graphs to represent complex networks do not provide a complete description of the real-world systems under investigation. For instance, WSN is undergoing intensive research to overcome its complexity and constraint challenges in terms of storage resources, computational capabilities, communication bandwidth, and more importantly, power supply.8 The main components of a sensor node and its associated energy consumption are discussed. Typically, sensor nodes are grouped hierarchically in clusters (sections), and each cluster has some nodes that acts as the cluster head (CH). All the nodes forward their sensor data to the CH, which in turn aggregates data reports and routes them to a specialized node called the sink node or base station (BS). A natural way of

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M. Hamidi & A. Borumand Saeid

representing these systems is to use hypergraphs. Hyper-edges in hypergraph can relate groups of more than two nodes. Thus, we can represent the collaboration network as a hypergraph in which nodes represent authors and hyper-edges represent the groups of authors who have published papers together. Despite the fact that complex weighted networks have been covered in some detail in the physical literature, there are no reports on the use of hypergraphs to represent complex systems.

Consequently, we will formally introduce the hypergraph concept as a generalization for representing complex networks and will call them complex hyper-networks. The hypergraph concept includes, as particular cases, a wide variety of other mathematical structures that are appropriate for the study of complex networks. Since these representations still are unsuccessful to deal with all the competitions of world, for that purpose SVN-HG are introduced. Now, we discuss applications of SVN-HG for studying the competition along with algorithms. The SVN-G has many utilizations in di®erent areas, where we by using the especial equivalence relations connect SVN-G and SVN-HG. We will show some examples of complex systems for which hypergraph representation is necessary.

Example 5.1. Let X ¼ fa; b; c; d; e; f; g; h; i; j; k; l; m; n; og be a WSN and a; b; c; d; e; f; g; h; i; j; k; l; m; n; o be nodes of its. These nodes create some groups as E1 ¼ fa; b; c; d; eg; E2 ¼ ff; g; h; ig, E3 ¼ fj; k; lg; E4 ¼ fm; ng and E5 ¼ fog. Let, the degree of remaining energy ðDREÞ in the WSN of a is 60=100, degree of sensor remaining alive ðDSRAÞ is 20=100 and degree of reliability ðDRÞ is 30=100, i.e. the truth-membership, indeterminacy-membership and falsity-membership values of the vertex human is ð0:6; 0:2; 0:3Þ. The degree of remaining energy, degree of sensor remaining alive and degree of reliability of WSN is shown in the Table 1.

Table 1. DRE, DSRA and DR of WS-HN. Node a b c d e f g h i j k l m n o

Truth-membership

Indeterminacy-membership

Falsity-membership

0.6 0.9 1 1 1 0.8 0.9 0.8 1 0.9 0.4 1 1 0.7 0.4

0.2 0.2 0.2 0.2 0.2 0.3 0.2 0.2 0.5 0.2 0.2 0.3 0.2 0.3 1

0.3 0.4 0.2 0.3 0.5 0.2 0.2 0.4 0.3 0.2 0.2 0.2 0.3 0.5 1

Achievable Single-Valued Neutrosophic Graphs in Wireless Sensor Networks

(i, 1, 0.5, 0.3) (h, 0.8, 0.2, 0.4) (g, 0.9, 0.2, 0.2) (f, 0.8, 0.3, 0.2)

(e, 1, 0.2, 0.5) (d, 1, 0.2, 0.3) (c, 1, 0.2, 0.2) (b, 0.9, 0.2, 0.4) (a, 0.6, 0.2, 0.3)

181

(l, 1, 0.3, 0.2) (k, 0.4, 0.2, 0.2) (j, 0.9, 0.2, 0.2)

(n, 0.7, 0.3, 0.5) (m, 1, 0.2, 0.3)

(o, 0.4, 1, 1)

Fig. 23. Clustering for wireless sensor hypernetwork (WS-HN).

Consider the clustering for wireless sensor hypernetwork H ¼ ðfa; b; c; d; e; f; g; h; i; j; k; l; m; n; og; fE1 ; E2 ; E3 ; E4 ; E5 gÞ in Fig. 23. Clearly ðða; 0:6; 0:2; 0:3ÞÞ ¼ fða; 0:6; 0:2; 0:5Þ; ðb; 0:6; 0:2; 0:5Þ; ðc; 0:6; 0:2; 0:5Þ; ðd; 0:6; 0:2; 0:5Þ; ðe; 0:6; 0:2; 0:5Þg; ððf; 0:8; 0:3; 0:2ÞÞ ¼ fðf; 0:8; 0:5; 0:4Þ; ðg; 0:8; 0:5; 0:4Þ; ðh; 0:8; 0:5; 0:4Þ; ði; 0:8; 0:5; 0:4Þg ððj; 0:9; 0:2; 0:2ÞÞ ¼ fðj; 0:4; 0:3; 0:2Þ; ðk; 0:4; 0:3; 0:2Þ; ðl; 0:4; 0:3; 0:2Þg; ððm; 1; 0:2; 0:3ÞÞ ¼ fðm; 0:7; 0:3; 0:5Þ; ðn; 0:7; 0:3; 0:5Þg and ððo; 0:4; 1; 1ÞÞ ¼ fðo; 0:4; 1; 1Þg: So we obtained the single-valued neutrosophic graph in Fig. 24, where 1 ¼ ð0:6; 0:5; 0:5Þ; 2 ¼ ð0:4; 0:5; 0:4Þ; 3 ¼ ð0:4; 0:3; 0:5Þ and 4 ¼ ð0:4; 1; 1Þ. By Fig. 24, for society X, we have 5 representatives ðaÞ; ðfÞ; ðjÞ; ðmÞ and ðoÞ where (η ∗ (a), 0.6, 0.2, 0.5) • λ1 • (η ∗ (f ), 0.8, 0.5, 0.4) (η ∗ (o), 0.4, 1, 1) • λ2 λ4 λ3 • (η ∗ (j), 0.4, 0.3, 0.2) (η ∗ (m), 0.7, 0.3, 0.5) • Fig. 24. Clustering of wireless sensor network (WSN).

182

M. Hamidi & A. Borumand Saeid Table 2. DRE, DSRA and DR of WSN. Cluster head ðaÞ ðfÞ ðjÞ ðmÞ

Cluster head

Truth

Indeterminacy

Falsity

ðfÞ ðjÞ ðmÞ ðoÞ

0.6 0.4 0.4 0.4

0.5 0.5 0.3 1

0.4 0.4 0.5 1

the likeness, indeterminacy and dislikeness of contribution in the business relationships of group of this society is equal share and is shown in Table 2.

Example 5.2. Let X ¼ fa; b; c; d; e; f; g; hg be a WSN and a; b; c; d; e; f; g; h be nodes. These nodes create some groups as E1 ¼ fa; bg; E2 ¼ fc; dg; E3 ¼ feg; E4 ¼ ffg and E5 ¼ fg; hg. Let, the degree of stability period and network lifetime ðDSPNLÞ of d is 30=100, degree of energy consumption and throughput (DECT) is 40=100 and degree of computational time ðDCTÞ is 50=100, i.e. the truthmembership, indeterminacy-membership and falsity-membership values of this node is ð0:3; 0:4; 0:5Þ. The degree of stability period and network lifetime, degree of energy consumption and throughput and computational time this WSN is shown in the Table 3.

Consider the energy of wireless sensor hyper-network is illustrated in Fig. 25.

Since ðða; 0:7; 0; 0:1ÞÞ ¼ fða; 0:7; 0:2; 0:3Þ; ðb; 0:7; 0:2; 0:3Þg; ððc; 0:4; 0:4; 0:4ÞÞ ¼ fðc; 0:3; 0:4; 0:5Þ; ðd; 0:3; 0:4; 0:5Þg ððe; 0; 0:1; 0:2ÞÞ ¼ fðe; 0; 0:1; 0:2Þg; ððf; 0:2; 0:3; 0:4ÞÞ ¼ fðf; 0:2; 0:3; 0:4Þg

and

ððg; 0:5; 0:5; 0:5ÞÞ ¼ fðg; 0:5; 0:7; 0:7Þ; ðh; 0:5; 0:7; 0:7Þg: So we obtained the single-valued neutrosophic graph in Fig. 26. By Fig. 26, for society X, we have 5 representatives ðaÞ; ðcÞ; ðeÞ; ðfÞ and ðgÞ where the likeness, indeterminacy and dislikeness of trophic relations between species of group of this society is shown in the Table 4.

Table 3. DSPNL, DECT and DCT of WS-HN. Node a b c d e f g h

Truth-membership

Indeterminacy-membership

Falsity-membership

0.7 0.8 0.4 0.3 0 0.2 0.5 0.6

0 0.2 0.4 0.4 0.1 0.3 0.5 0.7

0.1 0.3 0.4 0.5 0.2 0.4 0.5 0.7

183

(b, 0.8, 0.2, 0.3)

(a, 0.7, 0, 0.1)

(d, 0.3, 0.4, 0.5)

(c, 0.4, 0.4, 0.4)

(e, 0, 0.1, 0.2)

(f, 0.2, 0.3, 0.4)

(h, 0.6, 0.7, 0.7) (g, 0.5, 0.5, 0.5)

Achievable Single-Valued Neutrosophic Graphs in Wireless Sensor Networks

Fig. 25. Energy of wireless sensor hypernetwork (WS-HN).

(0 ,0 .4, 0.5 )

.5) 4, 0 , 0.

(0.2

η ∗ (f, 0.2, 0.3, 0.4) η ∗ (e, 0, 0.1, 0.2) • • (0 ,0 ) .7, 4 .

0 0.7 , ) 3 ) 0. 0.2, 0.3 ( , 0.2 .2 (0, , 0 0.7 ( , 0.

7) • • • η ∗ (a, 0.7, 0.2, 0.3) η ∗ (c, 0.3, 0.4, 0.5) η ∗ (g, 0.5, 0.7, 0.7)

.

Fig. 26. Energy of wireless sensor network (WSN).

Table 4. DSPNL, DECT and DCT of WSN. Cluster head ðfÞ ðfÞ ðfÞ ðeÞ ðeÞ ðeÞ

Cluster head

Truth

Indeterminacy

Falsity

ðaÞ ðcÞ ðgÞ ðaÞ ðcÞ ðgÞ

0.2 0.2 0.2 0 0 0

0.3 0.4 0.7 0.2 0.4 0.7

0.4 0.5 0.7 0.3 0.5 0.7

6. Conclusion The current paper considered the concept of single-valued neutrosophic hypergraphs as a generalization of single-valued neutrosophic hypergraphs via , as positive relation on single-valued neutrosophic hypergraphs. Moreover (i) It is considered single-valued neutrosophic hypergraphs corresponding to wireless sensor hypernetworks such that vertices ðV Þ represent the sensors and the set of links ðEÞ represents the connections between vertices.

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M. Hamidi & A. Borumand Saeid

(ii) Using the relation , is constructed G= as sensor clusters and via this quotient single-valued neutrosophic hypergraph the concept of (self) derivable single-valued neutrosophic graph and extended derivable single-valued neutrosophic graphs is introduced. (iii) It is obtained an equivalent condition that a single-valued neutrosophic graph is a non-derivable single-valued neutrosophic graph. (iv) The concept of intuitionistic neutrosophic sets provides an additional possibility to represent imprecise, uncertain, inconsistent and incomplete information which exist in real situations. In this research paper, we have described the concept of single-valued neutrosophic graphs. We have also presented applications of single-valued neutrosophic hypergraphs and single-valued neutrosophic graphs in wireless sensor network.

We hope that these results are helpful for further studies in single-valued neutrosophic graph theory. In our future studies, we hope to obtain more results regarding single-valued neutrosophic graphs, single-valued neutrosophic hypergraphs and their applications.

Acknowledgments We wish to thank the reviewers for excellent suggestions that have been incorporated into the paper.

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