Self-Centered Single Valued Neutrosophic Graphs

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 24 (2017) pp. 15536-15543 Š Research India Publications. http://www.ripublication.com

Self-Centered Single Valued Neutrosophic Graphs V.Krishnaraj Research Scholar, Research & Development Centre, Bharathiar University, Coimbatore - 641 046, India.

Orcid Id:0000-0001-8092-7524 R.Vikramaprasad Assistant Professor, Department of Mathematics, Government Arts College, Salem - 636 007, Tamil Nadu, India.

R.Dhavaseelan Assistant Professor, Department of Mathematics, Sona College of Technology, Salem - 636 005, Tamil Nadu, India.

Orcid Id:0000-0001-7035-4427

Abstract In this paper, we introduce the concepts of length, distance, eccentricity, radius, diameter, status, total status, median and central vertex of a single valued neutrosophic graph. We present the concept of self-centered single valued neutrosophic graph. We investigated some properties of selfcentered single valued neutrosophic graphs.

Keywords: Length; distance; eccentricity; radius; diameter; central vertex; status; median; self-centered single valued neutrosophic graph.

INTRODUCTION Fuzzy set [19] theory plays a vital role in complex phenomena which is not effortlessly described by classical set theory.

Atanassov introduced the concept of intuitionistic fuzzy relations and intuitionistic fuzzy graphs(IFGs). Parvathi and Karunambigai[13] introduced the concept of IFG elaborately and analyzed its components. Authors of [9] introduced the concept of self-centered IFG. Smarandache[6]-[7] introduced the idea of neutrosophic sets by combining the non-standard analysis. Neutrosophic set is a mathematical tool for dealing real life problems having imprecise, indeterminacy and inconsistent data. Neutrosophic set theory, as a generalization of classical set theory, fuzzy set theory and intuitionistic fuzzy set theory, is applied in a variety of fields, including control theory, decision making problems, topology, medicines and in many more real life problems. Wang et al.[16] presented the notion of single-valued neutrosophic sets to apply neutrosophic sets in real life problems more conveniently. A single-valued neutrosophic set has three components: truth membership degree, indeterminacy membership degree and falsity membership degree. These three components of a single-valued neutrosophic set are not dependent and their values are contained in the standard unit interval [0, 1].

Single-valued neutrosophic sets are the generalization of intuitionistic fuzzy sets. Single-valued neutrosophic sets have been a new hot research topic and many researchers have addressed this issue. Akram et al.[1-4] has discussed several concepts related to single-valued neutrosophic graphs.

Majumdar and Samanta [10] studied similarity and entropy of single-valued neutrosophic sets. Ye[18] proposed correlation coefficients of single-valued neutrosophic sets, and applied it to single-valued neutrosophic decision making problems.

In this paper, we introduce the concepts of length, distance, radius, eccentricity, diameter, status, total status, median and central vertex of a single valued neutrosophic graph. We present the concept of self-centered single valued neutrosophic graph. We also discuss some interesting properties besides giving some examples.

Definition 1.1 [17] Let X be a space of points. A neutrosophic set A in X is characterized by a truth-membership function đ?‘‡ đ??´ (đ?‘Ľ), an indeterminacy membership function đ??ź đ??´ (đ?‘Ľ) and a falsity membership function đ??š đ??´ (đ?‘Ľ) . The functions đ?‘‡ đ??´ (đ?‘Ľ), đ??ź đ??´ (đ?‘Ľ)and đ??š đ??´ (đ?‘Ľ) are real standard or non standard subsets of ]0− , 1+ [ . That is, đ?‘‡ đ??´ (đ?‘Ľ): đ?‘‹ →]0− , 1+ [ , đ??ź đ??´ (đ?‘Ľ): đ?‘‹ →]0− , 1+ [ , đ??š đ??´ (đ?‘Ľ): đ?‘‹ →]0− , 1+ [ and 0− ≤ đ?‘‡ đ??´ (đ?‘Ľ) + đ??ź đ??´ (đ?‘Ľ) + đ??š đ??´ (đ?‘Ľ) ≤ 3+ .

From philosophical point view, the neutrosophic set takes the value from real standard or non standard subsets of ]0− , 1+ [.

In real life applications in scientific and engineering problems, it is difficult to use neutrosophic set with value from real standard or non standard subset of ]0− , 1+ [.

Definition 1.2 [3, 1] A single valued neutrosophic graph is a pair đ??ş = (đ??´, đ??ľ) , where đ??´: đ?‘‰ → [0,1] is single valued neutrosophic set in V and đ??ľ: đ?‘‰ Ă— đ?‘‰ → [0,1] is single valued

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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 24 (2017) pp. 15536-15543 Š Research India Publications. http://www.ripublication.com neutrosophic relation on V such that đ?‘‡ đ??ľ (đ?‘Ľđ?‘Ś) ≤ min{đ?‘‡ đ??´ (đ?‘Ľ), đ?‘‡ đ??´ (đ?‘Ś)} , đ??ź đ??ľ (đ?‘Ľđ?‘Ś) ≤ min{đ??ź đ??´ (đ?‘Ľ), đ??ź đ??´ (đ?‘Ś)} , đ??š đ??ľ (đ?‘Ľđ?‘Ś) ≤ max{đ??š đ??´ (đ?‘Ľ), đ??š đ??´ (đ?‘Ś)} for all đ?‘Ľ, đ?‘Ś ∈ đ?‘‰ . A is called single valued neutrosophic vertex set of G and B is called single valued neutrosophic edge set of G, respectively.

We note that B is symmetric single valued neutrosophic relation on A. If B is not symmetric single valued neutrosophic relation on A, then đ??ş = (đ??´, đ??ľ) is called a single valued neutrosophic directed graph.

Definition 1.3 A single valued neutrosophic graph đ??ş = (đ??´, đ??ľ) is said to be complete if đ?‘‡đ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = min (đ?‘‡đ??´ (đ?‘Łđ?‘– ), đ?‘‡đ??´ (đ?‘Łđ?‘— )) , đ??źđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = min (đ??źđ??´ (đ?‘Łđ?‘– ), đ??źđ??´ (đ?‘Łđ?‘— ))and đ??šđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = max (đ??šđ??´ (đ?‘Łđ?‘– ), đ??šđ??´ (đ?‘Łđ?‘— )) , ∀đ?‘Łđ?‘– , đ?‘Łđ?‘— ∈ đ?‘‰.

SELF-CENTERED SINGLE VALUED NEUTROSOPHIC GRAPHS Definition 2.1 Let đ??ş = (đ??´, đ??ľ) be a single valued neutrosophic graph. Then the order of G is defined to be đ?‘‚(đ??ş) = (đ?‘‚đ?‘‡ (đ??ş), đ?‘‚đ??ź (đ??ş), đ?‘‚đ??š (đ??ş)) wheređ?‘‚ đ?‘‡ (đ??ş) = ∑đ?‘˘âˆˆđ?‘‰ đ?‘‡đ??´ (đ?‘˘), đ?‘‚ đ??ź (đ??ş) = ∑đ?‘˘âˆˆđ?‘‰ đ??źđ??´ (đ?‘˘),đ?‘‚đ??š (đ??ş) = ∑đ?‘˘âˆˆđ?‘‰ đ??šđ??´ (đ?‘˘).

Definition 2.2 The size of G is defined to be đ?‘†(đ??ş) = (đ?‘†đ?‘‡ (đ??ş), đ?‘†đ??ź (đ??ş), đ?‘†đ??š (đ??ş)) where đ?‘†đ?‘‡ (đ??ş) = ∑đ?‘˘,đ?‘Łâˆˆđ?‘‰ đ?‘‡đ??ľ (đ?‘˘, đ?‘Ł), đ?‘†đ??ź (đ??ş) = ∑đ?‘˘,đ?‘Łâˆˆđ?‘‰ đ??źđ??ľ (đ?‘˘, đ?‘Ł), đ?‘†đ??š (đ??ş) = ∑đ?‘˘,đ?‘Łâˆˆđ?‘‰ đ??šđ??ľ (đ?‘˘, đ?‘˘).

Definition 2.3 The neighbourhood of any vertex đ?‘Ł is defined as đ?‘ (đ?‘Ł) = (đ?‘ đ?‘‡ (đ?‘Ł), đ?‘ đ??ź (đ?‘Ł), đ?‘ đ??š (đ?‘Ł)) where đ?‘ đ?‘‡ (đ?‘Ł) = {đ?‘˘ ∈ đ?‘‰: đ?‘‡đ??ľ (đ?‘˘, đ?‘Ł) = min{đ?‘‡đ??´ (đ?‘˘), đ?‘‡đ??´ (đ?‘Ł)}}, đ?‘ đ??ź (đ?‘Ł) = {đ?‘˘ ∈ đ?‘‰: đ??źđ??ľ (đ?‘˘, đ?‘Ł) = min{đ??źđ??´ (đ?‘˘), đ??źđ??´ (đ?‘Ł)}}, đ?‘ đ??š (đ?‘Ł) = {đ?‘˘ ∈ đ?‘‰: đ??šđ??ľ (đ?‘˘, đ?‘Ł) = max{đ??šđ??´ (đ?‘˘), đ??šđ??´ (đ?‘Ł)}} andđ?‘ [đ?‘Ł] = đ?‘ (đ?‘Ł) âˆŞ {đ?‘Ł} is called closed neighbourhood of v.

Definition 2.4 A path P in a single valued neutrosophic graph đ??ş = (đ??´, đ??ľ) is a sequence of distinct vertices đ?‘Ł1 , đ?‘Ł2 , . . . , đ?‘Łđ?‘› such that either one of the following condition is satisfied (i) đ?‘‡đ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) > 0, đ??źđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) > 0andđ??šđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = 0 for some i and j. (ii) đ?‘‡đ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = 0, đ??źđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = 0 and đ??šđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) > 0 for some i and j.

Definition 2.5 Let G be a single valued neutrosophic graph. (i) [13]The length of a path đ?‘ƒ: đ?‘Ł1 , đ?‘Ł2 , . . . , đ?‘Łđ?‘›+1 (đ?‘› > 0) in G is n. (ii) [13]The path đ?‘ƒ: đ?‘Ł1 , đ?‘Ł2 , . . . , đ?‘Łđ?‘›+1 in G is called a cycle if đ?‘Ł1 = đ?‘Łđ?‘›+1 and đ?‘› ≼ 3 . (iii) An single valued neutrosophic graph G is connected if any two vertices are joined by path.

Definition 2.6 The strength of a path đ?‘ƒ: đ?‘Ł1 , đ?‘Ł2 , . . . , đ?‘Łđ?‘› , is defined as đ?‘†(đ?‘ƒ) = (đ?‘†đ?‘‡ (đ?‘ƒ), đ?‘†đ??ź (đ?‘ƒ), đ?‘†đ??š (đ?‘ƒ)) where, đ?‘†đ?‘‡ (đ?‘ƒ) =

min(đ?‘‡đ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— )), đ?‘†đ??ź (đ?‘ƒ) = min(đ??źđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— )) max(đ??šđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— )) for all i and j.

and

đ?‘†đ??š (đ?‘ƒ) =

Note 2.1 In other words, the strength of a path is defined to be the weight of the weakest edge of the path. i.e the strength of a path đ?‘†(đ?‘ƒ).

Definition 2.7 A single valued neutrosophic graph đ??ş = (đ??´, đ??ľ) is said to be a single valued neutrosophic bipartite if the vertex set V can be partitioned into two non empty sets đ?‘‰1 and đ?‘‰2 such that (i) đ?‘‡đ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = 0, đ??źđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = 0 and đ??šđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = 0 , if đ?‘Łđ?‘– , đ?‘Łđ?‘— ∈ đ?‘‰1 or đ?‘Łđ?‘– , đ?‘Łđ?‘— ∈ đ?‘‰2 , (ii) đ?‘‡đ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) > 0, đ??źđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) > 0 and đ??šđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) > 0, if đ?‘Łđ?‘– ∈ đ?‘‰1 or đ?‘Łđ?‘— ∈ đ?‘‰2 for some i and j (or) đ?‘‡đ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = 0, đ??źđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = 0 and đ??šđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) > 0 , if đ?‘Łđ?‘– ∈ đ?‘‰1 or đ?‘Łđ?‘— ∈ đ?‘‰2 for some i and j (or) đ?‘‡đ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) > 0, đ??źđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) > 0 and đ??šđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = 0, if đ?‘Łđ?‘– ∈ đ?‘‰1 or đ?‘Łđ?‘— ∈ đ?‘‰2 for some i and j.

Definition 2.8 A single valued neutrosophic bipartite graph đ??ş = (đ??´, đ??ľ) is said to be complete if đ?‘‡đ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = min(đ?‘‡đ??´ (đ?‘Łđ?‘– ), đ?‘‡đ??´ (đ?‘Łđ?‘— )), đ??źđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = min(đ??źđ??´ (đ?‘Łđ?‘– ), đ??źđ??´ (đ?‘Łđ?‘— )) and đ??šđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = max(đ??šđ??´ (đ?‘Łđ?‘– ), đ??šđ??´ (đ?‘Łđ?‘— )) for all đ?‘Łđ?‘– ∈ đ?‘‰1 and đ?‘Łđ?‘— ∈ đ?‘‰2 .

It is denoted by đ??žđ?‘‰1i ,đ?‘‰2i .

Definition 2.9 Let single valued neutrosophic graph đ??ť = (đ??´â€˛ , đ??ľâ€˛ ) is said to be a single valued neutrosophic subgraph of a connected single valued neutrosophic graph đ??ş = (đ??´, đ??ľ). If đ?‘‡đ??´â€˛ (đ?‘Łđ?‘– ) = đ?‘‡đ??´ (đ?‘Łđ?‘– ) , đ??źđ??´â€˛ (đ?‘Łđ?‘– ) = đ??źđ??´ (đ?‘Łđ?‘– ) , đ??šđ??´â€˛ (đ?‘Łđ?‘– ) = đ??šđ??´ (đ?‘Łđ?‘– )∀đ?‘Łđ?‘– ∈ đ?‘‰ ′ and đ?‘‡đ??ľâ€˛ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = đ?‘‡đ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) , đ??źđ??ľâ€˛ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = đ??źđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) , đ??šđ??ľâ€˛ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = đ??šđ??ľ (đ?‘Łđ?‘– , đ?‘Łđ?‘— )∀(đ?‘Łđ?‘– , đ?‘Łđ?‘— ) ∈ đ??¸ ′ .

Definition 2.10 Let đ??ş = (đ??´, đ??ľ) be a connected single valued neutrosophic graph. (i) The T-length of a path đ?‘ƒ: đ?‘Ł1 , đ?‘Ł2 , . . . , đ?‘Łđ?‘› in G, đ?‘™ đ?‘‡ (đ?‘ƒ) is defined as đ?‘™ đ?‘‡ (đ?‘ƒ) = ∑đ?‘›âˆ’1 đ?‘–=1 (

1

)

đ?‘‡đ??ľ (đ?‘Łđ?‘– ,đ?‘Łđ?‘–+1 )

(ii) The I-length of a path đ?‘ƒ: đ?‘Ł1 , đ?‘Ł2 , . . . , đ?‘Łđ?‘› in G, đ?‘™đ??ź (đ?‘ƒ) is 1

defined as đ?‘™đ??ź (đ?‘ƒ) = ∑đ?‘›âˆ’1 đ?‘–=1 (

)

đ??źđ??ľ (đ?‘Łđ?‘– ,đ?‘Łđ?‘–+1 )

(iii) The F-length of a path đ?‘ƒ: đ?‘Ł1 , đ?‘Ł2 , . . . , đ?‘Łđ?‘› in G, đ?‘™đ??š (đ?‘ƒ) is defined as đ?‘™đ??š (đ?‘ƒ) = ∑đ?‘›âˆ’1 đ?‘–=1 (

1

đ??šđ??ľ (đ?‘Łđ?‘– ,đ?‘Łđ?‘–+1 )

)

The (T,I,F)-length of a path đ?‘ƒ: đ?‘Ł1 , đ?‘Ł2 , . . . , đ?‘Łđ?‘› in G, đ?‘™(đ?‘‡,đ??ź,đ??š) (đ?‘ƒ) is defined as đ?‘™(đ?‘‡,đ??ź,đ??š) (đ?‘ƒ) = (đ?‘™ đ?‘‡ (đ?‘ƒ), đ?‘™đ??ź (đ?‘ƒ), đ?‘™đ??š (đ?‘ƒ)).

Definition 2.11 Let đ??ş = (đ??´, đ??ľ) be a connected single valued neutrosophic graph. (i) The T-distance đ?›żđ?‘‡ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) is the minimum of the T-length of all the paths joining đ?‘Łđ?‘– and đ?‘Łđ?‘— in G, where đ?‘Łđ?‘– , đ?‘Łđ?‘— ∈ đ?‘‰. i.e đ?›żđ?‘‡ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = min{đ?‘™ đ?‘‡ (đ?‘ƒ): P is a path between đ?‘Łđ?‘– and đ?‘Łđ?‘— },

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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 24 (2017) pp. 15536-15543 Š Research India Publications. http://www.ripublication.com (ii) The I-distance đ?›żđ??ź (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) is the minimum of the I-length of all the paths joining đ?‘Łđ?‘– and đ?‘Łđ?‘— in G, where đ?‘Łđ?‘– , đ?‘Łđ?‘— ∈ đ?‘‰ . i.eđ?›żđ??ź (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = min{đ?‘™đ??ź (đ?‘ƒ): P is a path between đ?‘Łđ?‘– and đ?‘Łđ?‘— }, (iii) The F-distance đ?›żđ??š (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) is the minimum of the F-length of all the paths joining đ?‘Łđ?‘– and đ?‘Łđ?‘— in G, where đ?‘Łđ?‘– , đ?‘Łđ?‘— ∈ đ?‘‰ . i.eđ?›żđ??š (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = min{đ?‘™đ??š (đ?‘ƒ): P is a path between đ?‘Łđ?‘– and đ?‘Łđ?‘— }, The distance đ?›ż(đ?‘‡,đ??ź,đ??š) (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) is defined as đ?›ż(đ?‘‡,đ??ź,đ??š) (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = (đ?›żđ?‘‡ , đ?›żđ??ź , đ?›żđ??š ).

Definition 2.12 Let đ??ş = (đ??´, đ??ľ) be a connected single valued neutrosophic graph. (i) For each đ?‘Łđ?‘– ∈ đ?‘‰ , the T-eccentricity of đ?‘Łđ?‘– , denoted by đ?‘’đ?‘‡ (đ?‘Łđ?‘– ) and is defined as đ?‘’đ?‘‡ (đ?‘Łđ?‘– ) = max{đ?›żđ?‘‡ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ): đ?‘Łđ?‘– ∈ đ?‘‰, đ?‘Łđ?‘– ≠đ?‘Łđ?‘— }. (ii) For each đ?‘Łđ?‘– ∈ đ?‘‰, the I-eccentricity of đ?‘Łđ?‘– , denoted by đ?‘’đ??ź (đ?‘Łđ?‘– ) and is defined as đ?‘’đ??ź (đ?‘Łđ?‘– ) = max{đ?›żđ??ź (đ?‘Łđ?‘– , đ?‘Łđ?‘— ): đ?‘Łđ?‘– ∈ đ?‘‰, đ?‘Łđ?‘– ≠đ?‘Łđ?‘— }. (iii) For each đ?‘Łđ?‘– ∈ đ?‘‰ , the F-eccentricity of đ?‘Łđ?‘– , denoted by đ?‘’đ??š (đ?‘Łđ?‘– ) and is defined as đ?‘’đ??š (đ?‘Łđ?‘– ) = min{đ?›żđ??š (đ?‘Łđ?‘– , đ?‘Łđ?‘— ): đ?‘Łđ?‘– ∈ đ?‘‰, đ?‘Łđ?‘– ≠đ?‘Łđ?‘— }.

For each đ?‘Łđ?‘– ∈ đ?‘‰, the eccentricity of đ?‘Łđ?‘– denoted by đ?‘’(đ?‘Łđ?‘– ) and is defined as đ?‘’(đ?‘Łđ?‘– ) = (đ?‘’đ?‘‡ (đ?‘Łđ?‘– ), đ?‘’đ??ź (đ?‘Łđ?‘– ), đ?‘’đ??š (đ?‘Łđ?‘– )).

Definition 2.13 Let đ??ş = (đ??´, đ??ľ) be a connected single valued neutrosophic graph. (i) The T-radius of G is denoted by đ?‘&#x;đ?‘‡ (đ??ş) and is defined as đ?‘&#x;đ?‘‡ (đ??ş) = min{đ?‘’đ?‘‡ (đ?‘Łđ?‘– ): đ?‘Łđ?‘– ∈ đ?‘‰}. (ii) The I-radius of G is denoted by đ?‘&#x;đ??ź (đ??ş) and is defined as đ?‘&#x;đ??ź (đ??ş) = min{đ?‘’đ??ź (đ?‘Łđ?‘– ): đ?‘Łđ?‘– ∈ đ?‘‰}. (iii) The F-radius of G is denoted by đ?‘&#x;đ??š (đ??ş) and is defined as đ?‘&#x;đ??š (đ??ş) = min{đ?‘’đ??š (đ?‘Łđ?‘– ): đ?‘Łđ?‘– ∈ đ?‘‰}.

The radius of G is denoted by đ?‘&#x;(đ??ş) and is defined as đ?‘&#x;(đ??ş) = (đ?‘&#x;đ?‘‡ (đ??ş), đ?‘&#x;đ??ź (đ??ş), đ?‘&#x;đ??š (đ??ş)).

Definition 2.14 Let đ??ş = (đ??´, đ??ľ) be a connected single valued neutrosophic graph. (i) The T-diameter of G is denoted by đ?‘‘đ?‘–đ?‘Ž đ?‘‡ (đ??ş) and is defined as đ?‘‘đ?‘–đ?‘Ž đ?‘‡ (đ??ş) = max{đ?‘’ đ?‘‡ (đ?‘Łđ?‘– ): đ?‘Łđ?‘– ∈ đ?‘‰}. (ii) The I-diameter of G is denoted by đ?‘‘đ?‘–đ?‘Ž đ??ź (đ??ş) and is defined as đ?‘‘đ?‘–đ?‘Ž đ??ź (đ??ş) = max{đ?‘’ đ??ź (đ?‘Łđ?‘– ): đ?‘Łđ?‘– ∈ đ?‘‰}. (iii) The F-diameter of G is denoted by đ?‘‘đ?‘–đ?‘Ž đ??š (đ??ş) and is defined as đ?‘‘đ?‘–đ?‘Ž đ??š (đ??ş) = max{đ?‘’ đ??š (đ?‘Łđ?‘– ): đ?‘Łđ?‘– ∈ đ?‘‰}.

The diameter of G is denoted by đ?‘‘đ?‘–đ?‘Ž(đ??ş) and is defined as đ?‘‘đ?‘–đ?‘Ž(đ??ş) = (đ?‘‘đ?‘–đ?‘Ž đ?‘‡ (đ??ş), đ?‘‘đ?‘–đ?‘Ž đ??ź (đ??ş), đ?‘‘đ?‘–đ?‘Ž đ??š (đ??ş)).

Example 2.1 Consider a single valued neutrosophic graph, đ??ş = (đ??´, đ??ľ) such that đ?‘‰ = {đ?‘Ł1 , đ?‘Ł2 , đ?‘Ł3 , đ?‘Ł4 , đ?‘Ł5 }, đ??¸ = {(đ?‘Ł1 , đ?‘Ł2 ), (đ?‘Ł2 , đ?‘Ł3 ), (đ?‘Ł3 , đ?‘Ł1 ), (đ?‘Ł3 , đ?‘Ł4 ), (đ?‘Ł4 , đ?‘Ł5 ), (đ?‘Ł5 , đ?‘Ł2 )}.

Path �

Distance (6,9,4) (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) (đ?‘‡,đ??ź,đ??š) Path

�

đ?‘Ł1 − đ?‘Ł2 đ?‘Ł1 − đ?‘Ł3 đ?‘Ł1 − đ?‘Ł4

đ?‘Ł1 − đ?‘Ł5

đ?‘Ł2 − đ?‘Ł3

(6,7,5) (11,12,10) (13,18,9) (8,8,8)

đ?‘Ł2 − đ?‘Ł4 đ?‘Ł2 − đ?‘Ł5 đ?‘Ł3 − đ?‘Ł4

Distance (13,13,10) (7,9,5) (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) (đ?‘‡,đ??ź,đ??š)

đ?‘Ł3 − đ?‘Ł5

đ?‘Ł4 − đ?‘Ł5

(5,5,5) (13,14,10) (8,9,5)

Then the eccentricity of đ?‘Łđ?‘– are đ?‘’(đ?‘Ł1 ) = (13,18,4) , đ?‘’(đ?‘Ł2 ) = (13,13,4) , đ?‘’(đ?‘Ł3 ) = (13,14,5) , đ?‘’(đ?‘Ł4 ) = (13,13,5) , đ?‘’(đ?‘Ł5 ) = (13,18,5). Radius of G is đ?‘&#x;(đ??ş) = (13,13,4) and Diameter of G is đ?‘‘(đ??ş) = (13,18,5).

Definition 2.15 A vertex đ?‘Łđ?‘– ∈ đ?‘‰ is called a (i) T-central vertex of a connected single valued neutrosophic graph G, if đ?‘&#x; đ?‘‡ (đ??ş) = đ?‘’ đ?‘‡ (đ?‘Łđ?‘– ). (ii) I-central vertex of a connected single valued neutrosophic graph G, ifđ?‘&#x; đ??ź (đ??ş) = đ?‘’ đ??ź (đ?‘Łđ?‘– ). (iii) F-central vertex of a connected single valued neutrosophic graph G, if đ?‘&#x; đ??š (đ??ş) = đ?‘’ đ??š (đ?‘Łđ?‘– ). (iv) Central vertex of a connected single valued neutrosophic graph G, if đ?‘&#x; đ?‘‡ (đ??ş) = đ?‘’ đ?‘‡ (đ?‘Łđ?‘– ) , đ?‘&#x; đ??ź (đ??ş) = đ?‘’ đ??ź (đ?‘Łđ?‘– ) and đ?‘&#x; đ??š (đ??ş) = đ?‘’ đ??š (đ?‘Łđ?‘– ) and the set of all central vertices of a single valued neutrosophic graph is denoted by đ??ś(đ??ş).

Definition 2.16 < đ??ś(đ??ş) >= đ??ť: (đ??´â&#x20AC;˛ , đ??ľâ&#x20AC;˛ ) is a single valued neutrosophic subgraph of đ??ş = (đ??´, đ??ľ) induced by the central vertices of G is called the center of G. Definition 2.17 Aconnected single valued neutrosophic graph G is a (i) T- self-centered single valued neutrosophic graph, if every vertex of G is a T- central vertex. (i.e) đ?&#x2018;&#x; đ?&#x2018;&#x2021; (đ??ş) = đ?&#x2018;&#x2019; đ?&#x2018;&#x2021; (đ?&#x2018;Łđ?&#x2018;&#x2013; ), â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;.

15538

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 24 (2017) pp. 15536-15543 Š Research India Publications. http://www.ripublication.com (ii) I- self-centered single valued neutrosophic graph, if every vertex of G is a I- central vertex. (i.e) đ?&#x2018;&#x; đ??ź (đ??ş) = đ?&#x2018;&#x2019; đ??ź (đ?&#x2018;Łđ?&#x2018;&#x2013; ), â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;. (iii) F- self-centered single valued neutrosophic graph, if every vertex of G is a F- central vertex. (i.e) đ?&#x2018;&#x; đ??š (đ??ş) = đ?&#x2018;&#x2019; đ??š (đ?&#x2018;Łđ?&#x2018;&#x2013; ), â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;. (iv) Single valued neutrosophic self-centered graph, if every vertex of G is a central vertex. (i.e) đ?&#x2018;&#x; đ?&#x2018;&#x2021; (đ??ş) = đ?&#x2018;&#x2019; đ?&#x2018;&#x2021; (đ?&#x2018;Łđ?&#x2018;&#x2013; ) , đ?&#x2018;&#x; đ??ź (đ??ş) = đ?&#x2018;&#x2019; đ??ź (đ?&#x2018;Łđ?&#x2018;&#x2013; ) and đ?&#x2018;&#x; đ??š (đ??ş) = đ?&#x2018;&#x2019; đ??š (đ?&#x2018;Łđ?&#x2018;&#x2013; ), â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;.

Example 2.2 Consider a single valued neutrosophic graph, đ??ş = (đ??´, đ??ľ) such that đ?&#x2018;&#x2030; = {đ?&#x2018;Ł1 , đ?&#x2018;Ł2 , đ?&#x2018;Ł3 , đ?&#x2018;Ł4 }, đ??¸ = {(đ?&#x2018;Ł1 , đ?&#x2018;Ł2 ), (đ?&#x2018;Ł2 , đ?&#x2018;Ł3 ), (đ?&#x2018;Ł3 , đ?&#x2018;Ł4 ), (đ?&#x2018;Ł4 , đ?&#x2018;Ł1 ), (đ?&#x2018;Ł1 , đ?&#x2018;Ł3 )}.

Path Distance đ?&#x203A;ż

(đ?&#x2018;&#x2021;,đ??ź,đ??š)

Path Distance đ?&#x203A;ż

(đ?&#x2018;&#x2021;,đ??ź,đ??š)

Definition 2.19 Let đ??ş = (đ??´, đ??ľ) be a connected single valued neutrosophic graph. (i) The minimum T-status of G is defined as đ?&#x2018;&#x161;[đ?&#x2018; đ?&#x2018;&#x2021; (đ??ş)] = đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;{đ?&#x2018; đ?&#x2018;&#x2021; (đ?&#x2018;˘): đ?&#x2018;˘ â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;}, (ii) The minimum I-status of G is defined as đ?&#x2018;&#x161;[đ?&#x2018; đ??ź (đ??ş)] = đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;{đ?&#x2018; đ??ź (đ?&#x2018;˘): đ?&#x2018;˘ â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;}, (iii) The minimum F-status of G is defined as đ?&#x2018;&#x161;[đ?&#x2018; đ??š (đ??ş)] = đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;{đ?&#x2018; đ??š (đ?&#x2018;˘): đ?&#x2018;˘ â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;}. (iv) The minimum status of G is denoted by đ?&#x2018;&#x161;[đ?&#x2018; (đ??ş)] and is defined as đ?&#x2018;&#x161;[đ?&#x2018; (đ??ş)] = (đ?&#x2018;&#x161;[đ?&#x2018; đ?&#x2018;&#x2021; (đ??ş)], đ?&#x2018;&#x161;[đ?&#x2018; đ??ź (đ??ş)], đ?&#x2018;&#x161;[đ?&#x2018; đ??š (đ??ş)]).

Definition 2.20 Let đ??ş = (đ??´, đ??ľ) be a connected single valued neutrosophic graph. (i) The maximum T-status of G is defined as đ?&#x2018;&#x20AC;[đ?&#x2018; đ?&#x2018;&#x2021; (đ??ş)] = đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;Ľ{đ?&#x2018; đ?&#x2018;&#x2021; (đ?&#x2018;˘): đ?&#x2018;˘ â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;}, (ii) The maximum I-status of G is defined as đ?&#x2018;&#x20AC;[đ?&#x2018; đ??ź (đ??ş)] = đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;Ľ{đ?&#x2018; đ??ź (đ?&#x2018;˘): đ?&#x2018;˘ â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;}, (iii) The maximum F-status of G is defined as đ?&#x2018;&#x20AC;[đ?&#x2018; đ??š (đ??ş)] = đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;Ľ{đ?&#x2018; đ??š (đ?&#x2018;˘): đ?&#x2018;˘ â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;}. (iv) The maximum status of G is denoted by đ?&#x2018;&#x20AC;[đ?&#x2018; (đ??ş)] and is defined as đ?&#x2018;&#x20AC;[đ?&#x2018; (đ??ş)] = (đ?&#x2018;&#x20AC;[đ?&#x2018; đ?&#x2018;&#x2021; (đ??ş)], đ?&#x2018;&#x20AC;[đ?&#x2018; đ??ź (đ??ş)], đ?&#x2018;&#x20AC;[đ?&#x2018; đ??š (đ??ş)]).

Definition 2.21 Let đ??ş = (đ??´, đ??ľ) be a connected single valued neutrosophic graph.

The total T-status of a node u of G is denoted by đ?&#x2018;Ąđ?&#x2018; đ?&#x2018;&#x2021; (đ?&#x2018;˘) and is defined as đ?&#x2018;Ąđ?&#x2018; đ?&#x2018;&#x2021; (đ?&#x2018;˘) = â&#x2C6;&#x2018;đ?&#x2018;˘â&#x2C6;&#x2C6;đ?&#x2018;&#x2030; đ?&#x2018; đ?&#x2018;&#x2021; (đ?&#x2018;˘), The total I-status of a node u of G is denoted by đ?&#x2018;Ąđ?&#x2018; đ??ź (đ?&#x2018;˘) and is defined as đ?&#x2018;Ąđ?&#x2018; đ??ź (đ?&#x2018;˘) = â&#x2C6;&#x2018;đ?&#x2018;Łâ&#x2C6;&#x2C6;đ?&#x2018;&#x2030; đ?&#x2018; đ??ź (đ?&#x2018;˘), The total F-status of a node u of G is denoted by đ?&#x2018;Ąđ?&#x2018; đ??š (đ?&#x2018;˘) and is defined as đ?&#x2018;Ąđ?&#x2018; đ??š (đ?&#x2018;˘) = â&#x2C6;&#x2018;đ?&#x2018;Łâ&#x2C6;&#x2C6;đ?&#x2018;&#x2030; đ?&#x2018; đ??š (đ?&#x2018;˘).

The total status of G is denoted by đ?&#x2018;Ą[đ?&#x2018; (đ??ş)] and is defined as đ?&#x2018;Ą[đ?&#x2018; (đ??ş)] = (đ?&#x2018;Ąđ?&#x2018; đ?&#x2018;&#x2021; (đ?&#x2018;˘), đ?&#x2018;Ąđ?&#x2018; đ??ź (đ?&#x2018;˘), đ?&#x2018;Ąđ?&#x2018; đ??š (đ?&#x2018;˘)).

đ?&#x2018;Ł1 â&#x2C6;&#x2019; đ?&#x2018;Ł2 đ?&#x2018;Ł1 â&#x2C6;&#x2019; đ?&#x2018;Ł3 đ?&#x2018;Ł1 â&#x2C6;&#x2019; đ?&#x2018;Ł4 (đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; ) (5,5,8) (11,11,12) (6,6,4) đ?&#x2018;Ł2 â&#x2C6;&#x2019; đ?&#x2018;Ł3 đ?&#x2018;Ł2 â&#x2C6;&#x2019; đ?&#x2018;Ł4 đ?&#x2018;Ł3 â&#x2C6;&#x2019; đ?&#x2018;Ł4 (đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; ) (6,6,4) (11,11,12) (5,5,8)

Then the eccentricity of đ?&#x2018;Łđ?&#x2018;&#x2013; are đ?&#x2018;&#x2019;(đ?&#x2018;Ł1 ) = (11,11,4) , đ?&#x2018;&#x2019;(đ?&#x2018;Ł2 ) = (11,11,4), đ?&#x2018;&#x2019;(đ?&#x2018;Ł3 ) = (11,11,4), đ?&#x2018;&#x2019;(đ?&#x2018;Ł4 ) = (11,11,4). Radius of G is đ?&#x2018;&#x;(đ??ş) = (11,11,4) and Diameter of G is đ?&#x2018;&#x2018;(đ??ş) = (11,11,4). Here đ?&#x2018;&#x;(đ??ş) = đ?&#x2018;&#x2019;(đ?&#x2018;Łđ?&#x2018;&#x2013; ), â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030; . Hence G is a selfcentered single valued neutrosophic graph.

Definition 2.18 Let đ??ş = (đ??´, đ??ľ) be a connected single valued neutrosophic graph. (i) The T-status of a node u of G is denoted by đ?&#x2018; đ?&#x2018;&#x2021; (đ?&#x2018;˘) and is defined as đ?&#x2018; đ?&#x2018;&#x2021; (đ?&#x2018;˘) = â&#x2C6;&#x2018;đ?&#x2018;Łâ&#x2C6;&#x2C6;đ?&#x2018;&#x2030; đ?&#x203A;żđ?&#x2018;&#x2021; (đ?&#x2018;˘, đ?&#x2018;Ł), (ii) The I-status of a node u of G is denoted by đ?&#x2018; đ??ź (đ?&#x2018;˘) and is defined as đ?&#x2018; đ??ź (đ?&#x2018;˘) = â&#x2C6;&#x2018;đ?&#x2018;Łâ&#x2C6;&#x2C6;đ?&#x2018;&#x2030; đ?&#x203A;żđ??ź (đ?&#x2018;˘, đ?&#x2018;Ł), (iii) The F-status of a node u of G is denoted by đ?&#x2018; đ??š (đ?&#x2018;˘) and is defined as đ?&#x2018; đ??š (đ?&#x2018;˘) = â&#x2C6;&#x2018;đ?&#x2018;Łâ&#x2C6;&#x2C6;đ?&#x2018;&#x2030; đ?&#x203A;żđ??š (đ?&#x2018;˘, đ?&#x2018;Ł), (iv) The status of a node u of G is defined as đ?&#x2018; (đ?&#x2018;˘) = (đ?&#x2018; đ?&#x2018;&#x2021; (đ?&#x2018;˘), đ?&#x2018; đ??ź (đ?&#x2018;˘), đ?&#x2018; đ??š (đ?&#x2018;˘)).

Definition 2.22 Let đ??ş = (đ??´, đ??ľ) be a connected single valued neutrosophic graph. The median is defined as đ?&#x2018;&#x20AC;(đ??ş) = (đ?&#x2018;&#x20AC;đ?&#x2018;&#x2021; (đ??ş), đ?&#x2018;&#x20AC;đ??ź (đ??ş), đ?&#x2018;&#x20AC;đ??š (đ??ş)) , where đ?&#x2018;&#x20AC;đ?&#x2018;&#x2021; (đ??ş) = {đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;: đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;{đ?&#x2018; đ?&#x2018;&#x2021; (đ?&#x2018;Łđ?&#x2018;&#x2013; )}} , đ?&#x2018;&#x20AC;đ??ź (đ??ş) = {đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;: đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;{đ?&#x2018; đ??ź (đ?&#x2018;Łđ?&#x2018;&#x2013; )}} , đ?&#x2018;&#x20AC;đ??š (đ??ş) = {đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;: đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;{đ?&#x2018; đ??š (đ?&#x2018;Łđ?&#x2018;&#x2013; )}}.

Example 2.3 Consider a single valued neutrosophic -graph, đ??ş = (đ??´, đ??ľ) such that đ?&#x2018;&#x2030; = {đ?&#x2018;Ł1 , đ?&#x2018;Ł2 , đ?&#x2018;Ł3 , đ?&#x2018;Ł4 }, đ??¸= {(đ?&#x2018;Ł1 , đ?&#x2018;Ł2 ), (đ?&#x2018;Ł2 , đ?&#x2018;Ł3 ), (đ?&#x2018;Ł3 , đ?&#x2018;Ł1 ), (đ?&#x2018;Ł3 , đ?&#x2018;Ł4 ), (đ?&#x2018;Ł1 , đ?&#x2018;Ł4 )}.

Here, status of the nodes are đ?&#x2018; (đ?&#x2018;Ł1 ) = (22,19,17), đ?&#x2018; (đ?&#x2018;Ł2 ) = (27,28,23), đ?&#x2018; (đ?&#x2018;Ł3 ) = (26,16,14), đ?&#x2018; (đ?&#x2018;Ł4 ) = (29,27,20).

The minimum status of G is đ?&#x2018;&#x161;[đ?&#x2018; (đ??ş)] = (22,16,14) . The maximum status of G is đ?&#x2018;&#x20AC;[đ?&#x2018; (đ??ş)] = (29,28,23) . The total status of G is đ?&#x2018;Ą[đ?&#x2018; (đ??ş)] = (104,90,74) . The median is đ?&#x2018;&#x20AC;(đ??ş) = ({đ?&#x2018;Ł1 }, {đ?&#x2018;Ł3 }, {đ?&#x2018;Ł3 }).

15539

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 24 (2017) pp. 15536-15543 Š Research India Publications. http://www.ripublication.com (đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2013;+1 ) is strong single valued neutrosophic for đ?&#x2018;&#x2013; = 1,2, . . . , đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 and the nodes are alternatively in đ?&#x2018;&#x2030;1 and đ?&#x2018;&#x2030;2 , we have đ?&#x2018;Łđ?&#x2018;&#x203A; and đ?&#x2018;Ł1 â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;1 . But this implies that (đ?&#x2018;Łđ?&#x2018;&#x203A; , đ?&#x2018;Ł1 ) is an edge in đ?&#x2018;&#x2030;1 which contradicts the assumption that G is a bipartite single valued neutrosophic graph. Hence bipartite single valued neutrosophic graph has no single valued neutrosophic strong cycle of odd length.

Theorem 2.2 Every complete single valued neutrosophic graph G is a self-centered single valued neutrosophic graph Definition 2.23 A connected single valued neutrosophic graph đ??ş = (đ??´, đ??ľ) is a self-median if all the nodes have the same status. In other words, a connected single valued neutrosophic graph đ??ş = (đ??´, đ??ľ) is self-median if and only if đ?&#x2018;&#x161;[đ?&#x2018; (đ??ş)] = đ?&#x2018;&#x20AC;[đ?&#x2018; (đ??ş)].

Example 2.4 Consider a single valued neutrosophic graph, đ??ş = (đ??´, đ??ľ) such that đ?&#x2018;&#x2030; = {đ?&#x2018;Ł1 , đ?&#x2018;Ł2 , đ?&#x2018;Ł3 , đ?&#x2018;Ł4 }, đ??¸= {(đ?&#x2018;Ł1 , đ?&#x2018;Ł2 ), (đ?&#x2018;Ł2 , đ?&#x2018;Ł3 ), (đ?&#x2018;Ł3 , đ?&#x2018;Ł4 ), (đ?&#x2018;Ł4 , đ?&#x2018;Ł1 ), (đ?&#x2018;Ł1 , đ?&#x2018;Ł3 ), (đ?&#x2018;Ł2 , đ?&#x2018;Ł4 )}.

Here, status of the nodes are (đ?&#x2018;Ł1 ) = (20,23,15), đ?&#x2018; (đ?&#x2018;Ł2 ) = (20,23,15), đ?&#x2018; (đ?&#x2018;Ł3 ) = (20,23,15), đ?&#x2018; (đ?&#x2018;Ł4 ) = (20,23,15) . The minimum status of G is đ?&#x2018;&#x161;[đ?&#x2018; (đ??ş)] = (20,23,15) . The maximum status of G is đ?&#x2018;&#x20AC;[đ?&#x2018; (đ??ş)] = (20,23,15) . The total status of G is đ?&#x2018;Ą[đ?&#x2018; (đ??ş)] = (80,92,60).

đ?&#x2018;&#x;(đ??ş) = (

and đ??ź

đ??´

1 đ?&#x2018;&#x2021;

đ??´

1

,

(đ?&#x2018;Ľ) đ??ź

đ??´

1

,

(đ?&#x2018;Ľ) đ??š

đ??´

(đ?&#x2018;Ľ) are the least value and đ??š

)

đ??´

đ?&#x2018;&#x2021;

where

(đ?&#x2018;Ľ)

đ??´

(đ?&#x2018;Ľ)

and

(đ?&#x2018;Ľ) is greatest value.

Proof. Let G be a complete single valued neutrosophic graph G. To prove that G is a self-centered single valued neutrosophic graph. That is we have to show that every vertex is a central vertex. First we claim that G is a T- self-centered single valued neutrosophic graph and đ?&#x2018;&#x; where đ?&#x2018;&#x2021; đ?&#x2018;&#x2021;

đ??´

đ??´

đ?&#x2018;&#x2021;

1

(đ??ş) =

đ?&#x2018;&#x2021;

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

,

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) is the least. Now fix a vertex đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030; such that

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) is least vertex membership value of G.

(1) Consider all the đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2019; đ?&#x2018;Łđ?&#x2018;&#x2014; paths P of length n in G, â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;.

Case (i) : If đ?&#x2018;&#x203A; = 1 , then đ?&#x2018;&#x2021; đ?&#x2018;&#x2021;

đ??´

đ??ľ

(đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; ) = min(đ?&#x2018;&#x2021;

(đ?&#x2018;Łđ?&#x2018;&#x2013; ). Therefore, the T-length of đ?&#x2018;&#x192; = đ?&#x2018;&#x2122;

đ?&#x2018;&#x2021;

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; )) = 1

(đ?&#x2018;&#x192;) =

đ?&#x2018;&#x2021;

đ??´

.

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

Case (ii) : If đ?&#x2018;&#x203A; > 1, then one of the edges of P possesses the T-strength đ?&#x2018;&#x2021; đ??´ (đ?&#x2018;Łđ?&#x2018;&#x2013; ) and hence, T-length of a đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2019; đ?&#x2018;Łđ?&#x2018;&#x2014; path will exceed

1 đ?&#x2018;&#x2021;

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

. That is T-length of P = đ?&#x2018;&#x2122;

đ?&#x2018;&#x2021;

(đ?&#x2018;&#x192;) >

1 đ?&#x2018;&#x2021;

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

.

Hence đ?&#x203A;ż

đ?&#x2018;&#x2021;

(đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; ) = min(đ?&#x2018;&#x2122;

đ?&#x2018;&#x2021;

1

(đ?&#x2018;&#x192;)) =

đ?&#x2018;&#x2021;

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

, â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2014; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;.

(1)

(2) Let đ?&#x2018;Łđ?&#x2018;&#x2DC; â&#x2030; đ?&#x2018;Łđ?&#x2018;&#x2013; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;&#x2030;. Consider all đ?&#x2018;Łđ?&#x2018;&#x2DC; â&#x2C6;&#x2019; đ?&#x2018;Łđ?&#x2018;&#x2014; paths Q of length n in G, â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2014; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;.

Case (i) : If đ?&#x2018;&#x203A; = 1, then đ?&#x2018;&#x2021; đ?&#x2018;&#x2021; đ?&#x2018;&#x2122; The median is đ?&#x2018;&#x20AC;(đ??ş) = {{đ?&#x2018;Ł1 , đ?&#x2018;Ł2 , đ?&#x2018;Ł3 , đ?&#x2018;Ł4 }, {đ?&#x2018;Ł1 , đ?&#x2018;Ł2 , đ?&#x2018;Ł3 , đ?&#x2018;Ł4 }, {đ?&#x2018;Ł1 , đ?&#x2018;Ł2 , đ?&#x2018;Ł3 , đ?&#x2018;Ł4 }}. Hence đ??ş = (đ??´, đ??ľ) is called the self-median graph.

Theorem 2.1 If đ??ş = (đ??´, đ??ľ) is a bipartite single valued neutrosophic graph then it has no strong single valued neutrosophic cycle of odd length.

Proof. Let G be a bipartite single valued neutrosophic graph with bipartition đ?&#x2018;&#x2030;1 and đ?&#x2018;&#x2030;2 . Suppose that it contains a strong cycle of odd length, say đ?&#x2018;Ł1 , đ?&#x2018;Ł2 , . . . , đ?&#x2018;Łđ?&#x2018;&#x203A; , đ?&#x2018;Ł1 for some odd n(vertices). Without loss of generality, let đ?&#x2018;Ł1 â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;1 . Since

đ??´ đ?&#x2018;&#x2021;

(đ?&#x2018;Łđ?&#x2018;&#x2013; ), since đ?&#x2018;&#x2021;

(đ?&#x2018;&#x201E;) =

Case

đ?&#x2018;&#x2021;

(ii) 1

đ?&#x2018;&#x2021;

đ??ľ

đ??´

1

(đ?&#x2018;Łđ?&#x2018;&#x2DC;+1 ,đ?&#x2018;Łđ?&#x2018;&#x2014; )

đ??ľ

(đ?&#x2018;Łđ?&#x2018;&#x2DC; ,đ?&#x2018;Łđ?&#x2018;&#x2014; )

(đ?&#x2018;Łđ?&#x2018;&#x2DC; , đ?&#x2018;Łđ?&#x2018;&#x2014; ) = min(đ?&#x2018;&#x2021;

â&#x2030;¤

2 đ?&#x2018;&#x2021;

đ??´

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2DC; , đ?&#x2018;Łđ?&#x2018;&#x2014; )) â&#x2030;Ľ

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) is the least. Hence T-length of đ?&#x2018;&#x201E; = 1 đ?&#x2018;&#x2021;

đ??´

.

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

If đ?&#x2018;&#x203A; = 2 ,

: â&#x2030;¤

đ??ľ

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

then đ?&#x2018;&#x2122;

, since đ?&#x2018;&#x2021;

Case (iii) : If đ?&#x2018;&#x203A; > 2, then đ?&#x2018;&#x2122;

đ?&#x2018;&#x2021;

đ??´

đ?&#x2018;&#x2021;

(đ?&#x2018;&#x201E;) =

1 đ?&#x2018;&#x2021;

đ??ľ

(đ?&#x2018;Łđ?&#x2018;&#x2DC; ,đ?&#x2018;Łđ?&#x2018;&#x2DC;+1 )

+

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) is the least.

(đ?&#x2018;&#x201E;) â&#x2030;¤

đ?&#x2018;&#x203A; đ?&#x2018;&#x2021;

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

, since đ?&#x2018;&#x2021;

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) is

the least. Hence đ?&#x203A;ż

đ?&#x2018;&#x2021;

(đ?&#x2018;Łđ?&#x2018;&#x2DC; , đ?&#x2018;Łđ?&#x2018;&#x2014; ) = min(đ?&#x2018;&#x2122;

đ?&#x2018;&#x2021;

(đ?&#x2018;&#x201E;)) â&#x2030;¤

1 đ?&#x2018;&#x2021;

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

, â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2DC; , đ?&#x2018;Łđ?&#x2018;&#x2014; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;.

(2)

, â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;.

(3)

From Equations (1) and (2), we have đ?&#x2018;&#x2019;

15540

đ?&#x2018;&#x2021;

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) = max(đ?&#x203A;ż

đ?&#x2018;&#x2021;

(đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; )) =

1 đ?&#x2018;&#x2021;

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 24 (2017) pp. 15536-15543 Š Research India Publications. http://www.ripublication.com Hence G is a T-self-centered single valued neutrosophic graph.

Now, đ?&#x2018;&#x; đ?&#x2018;&#x2021; (đ??ş) = min(đ?&#x2018;&#x2019; đ?&#x2018;&#x2021; (đ?&#x2018;Łđ?&#x2018;&#x2013; )) = đ?&#x2018;&#x;

(đ??ş) = đ?&#x2018;&#x2021;

1 đ?&#x2018;&#x2021;

đ??´

1 đ?&#x2018;&#x2021;

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

, đ?&#x2018;¤â&#x201E;&#x17D;đ?&#x2018;&#x2019;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x2021;

đ??š

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) đ?&#x2018;&#x2013;đ?&#x2018; đ?&#x2018;&#x2122;đ?&#x2018;&#x2019;đ?&#x2018;&#x17D;đ?&#x2018; đ?&#x2018;Ą. đ??´

Next, we claim that G is a I- self-centered single valued neutrosophic graph and đ?&#x2018;&#x; đ??ź (đ??ş) =

1 đ??ź

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

, where đ??ź

the least. Now fix a vertex đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030; such that đ??ź

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) is

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) is least đ??´

vertex membership value of G. (1) Consider all the đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2019; đ?&#x2018;Łđ?&#x2018;&#x2014; paths P of length n in G, â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;.

Case (i) : If đ?&#x2018;&#x203A; = 1 , then đ??ź đ??ź

đ??´

đ??ľ

(đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; ) = min(đ??ź

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) . Therefore, the I-length of đ?&#x2018;&#x192; = đ?&#x2018;&#x2122; đ??ź (đ?&#x2018;&#x192;) =

1 đ??ź

đ??´

1 đ??ź

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

1

. That is I-length of P = đ?&#x2018;&#x2122; đ??ź (đ?&#x2018;&#x192;) >

đ??ź

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

.

1

đ?&#x203A;ż đ??ź (đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; ) = min(đ?&#x2018;&#x2122; đ??ź (đ?&#x2018;&#x192;)) =

đ??ź

đ??´

, â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2014; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;.

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

(đ?&#x2018;Łđ?&#x2018;&#x2013; ), since đ??ź đ??´

đ?&#x2018;&#x2122; đ??ź (đ?&#x2018;&#x201E;) =

đ??ľ

(đ?&#x2018;Łđ?&#x2018;&#x2DC; ,đ?&#x2018;Łđ?&#x2018;&#x2014;)

â&#x2030;¤

1 đ??ź

đ??´

.

đ??ź

đ??´

, since đ??ź

1 đ??ź

đ??ľ

(đ?&#x2018;Łđ?&#x2018;&#x2DC; ,đ?&#x2018;Łđ?&#x2018;&#x2DC;+1 )

+

1 đ??ź

đ??ľ

(đ?&#x2018;Łđ?&#x2018;&#x2DC;+1 ,đ?&#x2018;Łđ?&#x2018;&#x2014; )

â&#x2030;¤

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) is the least. đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

Case (iii): If đ?&#x2018;&#x203A; > 2, then đ?&#x2018;&#x2122; đ??ź (đ?&#x2018;&#x201E;) â&#x2030;¤

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

đ??š

đ?&#x2018;&#x203A; đ??ź

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

, since đ??ź

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) is

1 đ??ź

đ?&#x2018;&#x2019; đ??ź (đ?&#x2018;Łđ?&#x2018;&#x2013; ) = max(đ?&#x203A;ż đ??ź (đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; )) =

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

, â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2DC; , đ?&#x2018;Łđ?&#x2018;&#x2014; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;.

đ?&#x2018;&#x; đ??ź (đ??ş) =

1 đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

đ??š

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; ), since đ??š

đ?&#x2018;&#x201E;=đ?&#x2018;&#x2122;

đ??š

(đ?&#x2018;&#x201E;) =

(ii):

1 đ??ź

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

1 , đ?&#x2018;¤â&#x201E;&#x17D;đ?&#x2018;&#x2019;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ??ź đ??ź đ??´ (đ?&#x2018;Łđ?&#x2018;&#x2013; )

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

.

đ??š

(đ?&#x2018;&#x192;) >

1 đ??š

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

.

đ??š

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

, â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2014; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;.

(7)

đ??ľ

đ??´

đ??š

đ??ľ

â&#x2030;Ľ

1 đ??š

đ??´

đ?&#x2018;&#x203A;=2 , đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2DC; , đ?&#x2018;Łđ?&#x2018;&#x2014; ) = max(đ??š

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2DC; , đ?&#x2018;Łđ?&#x2018;&#x2014; )) â&#x2030;¤

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) is the greatest. Hence F-length of

(đ?&#x2018;Łđ?&#x2018;&#x2DC; ,đ?&#x2018;Łđ?&#x2018;&#x2014; )

2 đ??š

đ??ľ

. đ?&#x2018;&#x2122;

then

, since đ??š

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

Case (iii): If đ?&#x2018;&#x203A; > 2, then đ?&#x2018;&#x2122;

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

đ??š

đ??´

đ??š

(đ?&#x2018;&#x201E;) =

1 đ??š

đ??ľ

(đ?&#x2018;Łđ?&#x2018;&#x2DC; ,đ?&#x2018;Łđ?&#x2018;&#x2DC;+1 )

+

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) is the greatest. đ?&#x2018;&#x203A;

(đ?&#x2018;&#x201E;) â&#x2030;Ľ

đ??š

đ??´

, since đ??š

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) is

the greatest. Hence đ?&#x203A;ż

đ??š

(đ?&#x2018;Łđ?&#x2018;&#x2DC; , đ?&#x2018;Łđ?&#x2018;&#x2014; ) = min(đ?&#x2018;&#x2122;

đ??š

1

(đ?&#x2018;&#x201E;)) â&#x2030;Ľ

đ??š

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

, â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2DC; , đ?&#x2018;Łđ?&#x2018;&#x2014; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;.

(8)

đ?&#x2018;&#x2019;

đ??š

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) = min(đ?&#x203A;ż

, â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;.

(đ??ş) =

(6)

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

, where đ??š

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; )) =

1 đ??š

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

, â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;.

(9)

Hence G is a F-self-centered single valued neutrosophic graph.

Now, đ?&#x2018;&#x; đ??š (đ??ş) = min(đ?&#x2018;&#x2019; đ??š (đ?&#x2018;Łđ?&#x2018;&#x2013; ))

đ??š

(đ??ş) =

1 đ??š

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

1 đ??š

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

, đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;?đ?&#x2018;&#x2019;đ?&#x2018;?đ?&#x2018;Śđ?&#x2018;&#x2019;đ?&#x2018;&#x17E;đ?&#x2018;˘đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A; (9) , đ?&#x2018;¤â&#x201E;&#x17D;đ?&#x2018;&#x2019;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ??š

đ??´

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) đ?&#x2018;&#x2013;đ?&#x2018; đ?&#x2018;&#x201D;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x2019;đ?&#x2018; đ?&#x2018;Ą.

Theorem 2.3 A single valued neutrosophic graph đ??ş = (đ??´, đ??ľ) is a self-centered single valued neutrosophic graph iff đ?&#x203A;ż đ?&#x2018;&#x2021; (đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; ) â&#x2030;¤ đ?&#x2018;&#x; đ?&#x2018;&#x2021; (đ??ş) , đ?&#x203A;ż đ??ź (đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; ) â&#x2030;¤ đ?&#x2018;&#x; đ??ź (đ??ş) and đ?&#x203A;ż

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) đ?&#x2018;&#x2013;đ?&#x2018; đ?&#x2018;&#x2122;đ?&#x2018;&#x2019;đ?&#x2018;&#x17D;đ?&#x2018; đ?&#x2018;Ą. đ??´ 1

đ??š

đ??š

From equations (3),(6), and (9), every vertex of G is a central vertex. Hence G is a self-centered single valued neutrosophic graph.

Next, we claim that G is a F - self-centered single valued đ??š

1

(đ?&#x2018;&#x192;)) =

1

If â&#x2030;Ľ

(đ?&#x2018;Łđ?&#x2018;&#x2DC;+1 ,đ?&#x2018;Łđ?&#x2018;&#x2014;)

đ?&#x2018;&#x;

, đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;?đ?&#x2018;&#x2019;đ?&#x2018;?đ?&#x2018;Śđ?&#x2018;&#x2019;đ?&#x2018;&#x17E;đ?&#x2018;˘đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A; (6)

neutrosophic graph and đ?&#x2018;&#x;

đ??š

Case (i): If đ?&#x2018;&#x203A; = 1, then đ??š

(5)

Hence G is a I-self-centered single valued neutrosophic graph.

Now, đ?&#x2018;&#x; đ??ź (đ??ş) = min(đ?&#x2018;&#x2019; đ??ź (đ?&#x2018;Łđ?&#x2018;&#x2013; )) đ??ź

1 đ??š

(2) Let đ?&#x2018;Łđ?&#x2018;&#x2DC; â&#x2030; đ?&#x2018;Łđ?&#x2018;&#x2013; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;&#x2030;. Consider all đ?&#x2018;Łđ?&#x2018;&#x2DC; â&#x2C6;&#x2019; đ?&#x2018;Łđ?&#x2018;&#x2014; paths Q of length n

=

From Equations (4) and (5), we have

=

. That is F-length of P= đ?&#x2018;&#x2122;

(đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; ) = min(đ?&#x2018;&#x2122;

the least. Hence đ?&#x203A;ż đ??ź (đ?&#x2018;Łđ?&#x2018;&#x2DC; , đ?&#x2018;Łđ?&#x2018;&#x2014; ) = min(đ?&#x2018;&#x2122; đ??ź (đ?&#x2018;&#x201E;)) â&#x2030;¤

(đ?&#x2018;&#x192;) =

From Equations (7) and (8), we have

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

Case (ii): If đ?&#x2018;&#x203A; = 2, then đ?&#x2018;&#x2122; đ??ź (đ?&#x2018;&#x201E;) = 2

(đ?&#x2018;Łđ?&#x2018;&#x2DC; , đ?&#x2018;Łđ?&#x2018;&#x2014; )) â&#x2030;Ľ

đ??´

đ??ź

1

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) is the least. Hence I-length of đ?&#x2018;&#x201E; = đ??´

1 đ??ź

(đ?&#x2018;Łđ?&#x2018;&#x2DC; , đ?&#x2018;Łđ?&#x2018;&#x2014; ) = min(đ??ź

đ??š

(đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; )) =

đ??´

Hence

đ??š

in G, â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2014; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;. đ??ľ

1

exceed

(4)

(2) Let đ?&#x2018;Łđ?&#x2018;&#x2DC; â&#x2030; đ?&#x2018;Łđ?&#x2018;&#x2013; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;&#x2030;. Consider all đ?&#x2018;Łđ?&#x2018;&#x2DC; â&#x2C6;&#x2019; đ?&#x2018;Łđ?&#x2018;&#x2014; paths Q of length n Case (i) : If đ?&#x2018;&#x203A; = 1 , then đ??ź

(đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; ) = max(đ??š

(đ?&#x2018;Łđ?&#x2018;&#x2013; ). Therefore, the F - length of đ?&#x2018;&#x192; = đ?&#x2018;&#x2122;

Case

Hence

đ??ź

đ??´

đ??ľ

in G, â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2014; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;.

.

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

Case (ii) : If đ?&#x2018;&#x203A; > 1, then one of the edges of P possesses the Istrength đ??ź đ??´ (đ?&#x2018;Łđ?&#x2018;&#x2013; ) and hence, I-length of a đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2019; đ?&#x2018;Łđ?&#x2018;&#x2014; path will exceed

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) is

Case (ii): If đ?&#x2018;&#x203A; > 1, then one of the edges of P possesses the Fstrength đ??ź đ??´ (đ?&#x2018;Łđ?&#x2018;&#x2013; ) and hence, F-length of a đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2019; đ?&#x2018;Łđ?&#x2018;&#x2014; path will

đ?&#x203A;ż

(đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; )) =

đ??´

greatest vertex membership value of G. (1) Consider all the đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2019; đ?&#x2018;Łđ?&#x2018;&#x2014; paths P of length n in G, â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;.

Case (i): If đ?&#x2018;&#x203A; = 1 , then đ??š

, đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;?đ?&#x2018;&#x2019;đ?&#x2018;?đ?&#x2018;Śđ?&#x2018;&#x2019;đ?&#x2018;&#x17E;đ?&#x2018;˘đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A;(3)

(đ?&#x2018;Łđ?&#x2018;&#x2013; )

the greatest. Now fix a vertex đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030; such that đ??š

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) is

đ??š

(đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; ) â&#x2030;Ľ đ?&#x2018;&#x;

đ??š

(đ??ş)â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;.

Proof.â&#x2021;&#x2019; We assume that G is self-centered single valued neutrosophic graph G. That is đ?&#x2018;&#x2019; đ?&#x2018;&#x2021; (đ?&#x2018;Łđ?&#x2018;&#x2013; ) = đ?&#x2018;&#x2019; đ?&#x2018;&#x2021; (đ?&#x2018;Łđ?&#x2018;&#x2014; ), đ?&#x2018;&#x2019; đ??ź (đ?&#x2018;Łđ?&#x2018;&#x2013; ) = đ?&#x2018;&#x2019; đ??ź (đ?&#x2018;Łđ?&#x2018;&#x2014; ), đ?&#x2018;&#x2019; đ?&#x2018;&#x2019;

đ?&#x2018;&#x2021;

(đ?&#x2018;Łđ?&#x2018;&#x2013; ) = đ?&#x2018;&#x2019;

đ??š

(đ?&#x2018;Łđ?&#x2018;&#x2014; ), â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030;, đ?&#x2018;&#x;

(đ?&#x2018;Łđ?&#x2018;&#x2013; ), đ?&#x2018;&#x; đ??ź (đ??ş) = đ?&#x2018;&#x2019; đ??ź (đ?&#x2018;Łđ?&#x2018;&#x2013; ) and đ?&#x2018;&#x;

Now

15541

đ??š

we

wish

to

đ??š

show

đ?&#x2018;&#x2021;

(đ??ş) = đ?&#x2018;&#x2019; that

(đ??ş) = đ??š

(đ?&#x2018;Łđ?&#x2018;&#x2013; ), â&#x2C6;&#x20AC;đ?&#x2018;Łđ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2018;&#x2030; . đ?&#x203A;ż

đ?&#x2018;&#x2021;

(đ?&#x2018;Łđ?&#x2018;&#x2013; , đ?&#x2018;Łđ?&#x2018;&#x2014; ) â&#x2030;¤

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 24 (2017) pp. 15536-15543 © Research India Publications. http://www.ripublication.com 𝑟

𝑇

(𝐺), 𝛿 𝐼 (𝑣𝑖 , 𝑣𝑗 ) ≤ 𝑟

(𝑣𝑖 , 𝑣𝑗 ) ≥

𝑇

𝑟

𝐹

(𝐺), ∀𝑣𝑖 , 𝑣𝑗 ∈ 𝑉 . By the definition of eccentricity, we

𝐼

𝛿

obtain,

𝑇

(𝐺)

(𝑣𝑖 , 𝑣𝑗 ) ≤ 𝑒 𝑇

𝛿

and

𝐹

(𝑣𝑖 ), 𝛿 𝐼 (𝑣𝑖 , 𝑣𝑗 ) ≤ 𝑒 𝐼 (𝑣𝑖 ) 𝑇

𝛿

(𝑣𝑖 , 𝑣𝑗 ) ≥ 𝑒 𝐹

(𝑣𝑖 ), ∀𝑣𝑖 , 𝑣𝑗 ∈ 𝑉. 𝐹

𝑒

(𝑣𝑗 ), 𝑒 𝐼 (𝑣𝑖 ) = 𝑒 𝐼 (𝑣𝑗 ) , 𝑒 𝑇

𝑒

When

(𝑣𝑖 ) = 𝑇

𝐹 𝑇

Since G is self-centered single valued neutrosophic graph, the above inequality becomes 𝛿 𝑇 (𝑣𝑖 , 𝑣𝑗 ) ≤

𝐼

(𝐺), 𝛿 𝐼 (𝑣𝑖 , 𝑣𝑗 ) ≤ 𝑟

𝑇

⇐ Assume that 𝛿 and 𝛿

𝐹

(𝑣𝑖 , 𝑣𝑗 ) ≥ 𝑟

𝑇

𝑇

(𝐺) and 𝛿

(𝑣𝑖 , 𝑣𝑗 ) ≤ 𝑟 𝐹

𝑇

𝐹

(𝑣𝑖 , 𝑣𝑗 ) ≥ 𝑟

𝐹

(𝐺).

(𝐺), 𝛿 𝐼 (𝑣𝑖 , 𝑣𝑗 ) ≤ 𝑟 𝐼 (𝐺)

(𝐺), ∀𝑣𝑖 , 𝑣𝑗 ∈ 𝑉.Then we have to prove

that G is self-centered single valued neutrosophic graph.

Suppose that G is not self-centered single valued neutrosophic graph.

Then 𝑟 𝑇 (𝐺) = 𝑒 𝑇 (𝑣𝑖 ), 𝑟 𝐼 (𝐺) = 𝑒 𝐼 (𝑣𝑖 ) and𝑟 𝑒

𝐹

(𝐺) = 𝑒

𝐹

(𝑣𝑖 ), for some𝑣𝑖 ∈ 𝑉. Let us assume that

(𝑣𝑖 ), 𝑒 𝐼 (𝑣𝑖 ) and 𝑒

𝑇

𝐹

(𝑣𝑖 ) is the least value among all

other eccentricity. That is 𝑟 𝑇 (𝐺) = 𝑒 𝑇 (𝑣𝑖 ), 𝑟 𝐼 (𝐺) = 𝑒 𝐼 (𝑣𝑖 ) 𝑎𝑛𝑑𝑟 𝑒

𝐹

(𝐺) =

(𝑣𝑖 )

𝐹

Where

(10) 𝑒

𝑇

(𝑣𝑖 ) < 𝑒

𝑇

(𝑣𝑗 ), 𝑒 𝐼 (𝑣𝑖 ) < 𝑒 𝐼 (𝑣𝑗 ), 𝑒

𝑒

𝐹

(𝑣𝑗 ), for some 𝑣𝑖 , 𝑣𝑗 ∈ 𝑉 and

𝛿

𝑇

(𝑣𝑖 , 𝑣𝑗 ) = 𝑒

𝑇

(𝑣𝑗 ) > 𝑒

𝑇

𝐹

(𝑣𝑖 ) <

(𝑣𝑖 ), 𝛿 𝐼 (𝑣𝑖 , 𝑣𝑗 ) = 𝑒 𝐼 (𝑣𝑗 ) >

𝑒 𝐼 (𝑣𝑖 ) 𝑎𝑛𝑑 𝛿

𝐹

(𝑣𝑖 , 𝑣𝑗 ) = 𝑒

(𝑣𝑗 ) > 𝑒

𝐹

𝑟

(𝐺), 𝛿 𝐼 (𝑣𝑖 , 𝑣𝑗 ) > 𝑟 𝐼 (𝐺) and 𝛿 𝑇

𝑇

(𝑣𝑖 , 𝑣𝑗 ) < 𝑟 𝐹

(𝑣𝑖 , 𝑣𝑗 ) > (𝐺), for 𝐹

𝛿

𝑇

(𝑣𝑖 , 𝑣𝑗 ) ≤ 𝑟

𝑇

(𝐺), 𝛿 𝐼 (𝑣𝑖 , 𝑣𝑗 ) ≤ 𝑟

𝐹

(𝑣𝑖 , 𝑣𝑗 ) ≥ 𝑟

𝐹

(𝐺), ∀𝑣𝑖 , 𝑣𝑗 ∈ 𝑉. Hence G is a self-centered

𝑇

(𝐺)

and

single valued neutrosophic graph.

(𝑣𝑖 , 𝑣𝑗 ) = max(𝐹

𝐹

𝐵 𝐵

𝐼

𝐵

(𝑣𝑖 , 𝑣𝑗 ) = min (𝐼

𝐹

𝐵

(𝑣𝑖 , 𝑣𝑗 ) = max (𝐹

(𝑣𝑖 ), 𝑇 𝐴 𝐴

(𝑣𝑖 ), 𝐼

𝐴

(𝑣𝑖 ), 𝐹

(𝑣𝑗 )), 𝐴 𝐴

𝐵

𝐼 𝐹

𝐵

(𝑣𝑖 , 𝑣𝑗 ) = 0,

𝐵

(𝑣𝑖 , 𝑣𝑗 ) = 0, ∀𝑣𝑖 , 𝑣𝑗 ∈ 𝑉1 (𝑜𝑟) 𝑣𝑖 , 𝑣𝑗 ∈ 𝑉2

(𝑣𝑗 )) − 𝑇

(𝑣𝑗 )) − 𝐼

𝐴

(𝑣𝑖 ), 𝐹

𝐴

𝐵

𝐵

(𝑣𝑖 , 𝑣𝑗 )

(13)

(𝑣𝑖 , 𝑣𝑗 )

(𝑣𝑗 )) − 𝐹

𝐵

(𝑣𝑖 , 𝑣𝑗 ).

(𝑣𝑖 ), 𝑇

(𝑣𝑖 ), 𝐼 𝐴

𝐴

(𝑣𝑗 ))

(14)

(𝑣𝑗 ))

𝐴

(𝑣𝑖 ), 𝐹

𝐴

(15)

(𝑣𝑗 )),

∀𝑣𝑖 , 𝑣𝑗 ∈ 𝑉1 (𝑜𝑟) 𝑣𝑖 , 𝑣𝑗 ∈ 𝑉2

(16)

From equations (12),(14), the complement of G has two components and each is complete single valued neutrosophic graph, which are self-centered single valued neutrosophic by Theorem 2.2.Hence the proof.

Theorem 2.5 Every self-median SVN-graph is a selfcentered SVN-graph.

Proof. Let 𝐺 = (𝐴, 𝐵) be a connected self-median SVN-graph with 𝑉 = {𝑣1 , 𝑣2 , 𝑣3 , . . . , 𝑣𝑛 }.

By definition, 𝑠𝑇 (𝑣1 ) = 𝑠𝑇 (𝑣2 ) = 𝑠𝑇 (𝑣3 ) =. . . = 𝑠𝑇 (𝑣𝑛 ), 𝑠𝐼 (𝑣1 ) = 𝑠𝐼 (𝑣2 ) = 𝑠𝐼 (𝑣3 ) =. . . = 𝑠𝐼 (𝑣𝑛 ), 𝑠𝐹 (𝑣1 ) = 𝑠𝐹 (𝑣2 ) = 𝑠𝐹 (𝑣3 ) =. . . = 𝑠𝐹 (𝑣𝑛 ). ∑𝑣𝑖∈𝑉 𝛿𝑇 (𝑣1 , 𝑣𝑖 ) = ∑𝑣𝑖 ∈𝑉 𝛿𝑇 (𝑣2 , 𝑣𝑖 ) = ∑𝑣𝑖∈𝑉 𝛿𝑇 (𝑣3 , 𝑣𝑖 ) = 𝑖≠2

𝑖≠3

∑𝑣𝑖∈𝑉 𝛿𝐼 (𝑣1 , 𝑣𝑖 ) = ∑𝑣𝑖∈𝑉 𝛿𝐼 (𝑣2 , 𝑣𝑖 ) = ∑𝑣𝑖∈𝑉 𝛿𝐼 (𝑣3 , 𝑣𝑖 ) =. . . = 𝑖≠1

𝑖≠2

𝑖≠𝑛

∑𝑣𝑖∈𝑉 𝛿𝐹 (𝑣1 , 𝑣𝑖 ) = ∑𝑣𝑖∈𝑉 𝛿𝐹 (𝑣2 , 𝑣𝑖 ) = ∑𝑣𝑖∈𝑉 𝛿𝐹 (𝑣3 , 𝑣𝑖 ) = 𝑖≠1

𝑖≠2

𝑖≠3

. . . = ∑𝑣𝑖 ∈𝑉 𝛿𝐹 (𝑣𝑛 , 𝑣𝑖 ). 𝑖≠𝑛 1

1

∑𝑣𝑖∈𝑉

𝑇𝐵 (𝑣𝑛 ,𝑣𝑖 )

𝑖≠𝑛

1

∑𝑣𝑖∈𝑉

𝐼𝐵 (𝑣𝑛 ,𝑣𝑖 )

𝑖≠𝑛

𝐼𝐵 (𝑣2 ,𝑣𝑖 )

𝑖≠2

𝑇𝐵 (𝑣3 ,𝑣𝑖 )

𝑖≠3

=. . . =

= ∑𝑣𝑖∈𝑉 𝑖≠3

1 𝐼𝐵 (𝑣3 ,𝑣𝑖 )

=. . . =

, 1

= ∑𝑣𝑖∈𝑉

𝐹𝐵 (𝑣1 ,𝑣𝑖 ) 1

1

= ∑𝑣𝑖∈𝑉

1

∑𝑣𝑖∈𝑉

1

= ∑𝑣𝑖∈𝑉

,

𝐼𝐵 (𝑣1 ,𝑣𝑖 )

𝑖≠1

𝑇𝐵 (𝑣2 ,𝑣𝑖 )

𝑖≠2

1

∑𝑣𝑖∈𝑉

1

= ∑𝑣𝑖∈𝑉

𝑇𝐵 (𝑣1 ,𝑣𝑖 )

𝐹𝐵 (𝑣2 ,𝑣𝑖 )

𝑖≠2

1

= ∑𝑣𝑖 ∈𝑉

𝐹𝐵 (𝑣3 ,𝑣𝑖 )

𝑖≠3

=. . . =

.

𝐹𝐵 (𝑣𝑛 ,𝑣𝑖 )

𝑖≠𝑛

𝑚𝑎𝑥{

1

} = 𝑚𝑎𝑥{

𝑇𝐵 (𝑣1 ,𝑣𝑖 ) 1

𝑚𝑎𝑥{ (12)

𝑖≠3

∑𝑣𝑖∈𝑉 𝛿𝐼 (𝑣𝑛 , 𝑣𝑖 ),

∑𝑣𝑖∈𝑉

(𝑣𝑗 )), ∀𝑣𝑖 ∈ 𝑉1 , 𝑣𝑗 ∈ 𝑉2 .

And 𝑇 𝐵 (𝑣𝑖 , 𝑣𝑗 ) = 0,

𝐴

(𝑣𝑖 , 𝑣𝑗 ) = max (𝐹

𝑖≠1

(𝑣𝑗 )), 𝐴

𝐴

(𝑣𝑖 , 𝑣𝑗 ) = min(𝐼

∑𝑣𝑖∈𝑉

Theorem 2.4 Let 𝐺 = (𝐴, 𝐵) be a single valued neutrosophic graph. If the graph G is complete bipartite single valued neutrosophic graph then the complement of G is self-centered single valued neutrosophic graph.

Proof. A bipartite single valued neutrosophic graph G is said to be complete, if (𝑣𝑖 , 𝑣𝑗 ) = min (𝑇 𝐵

𝐴

(𝑣𝑖 , 𝑣𝑗 ) = min(𝑇

𝑖≠1

𝑇

(𝑣𝑖 ), 𝐼

𝐴

𝑖≠𝑛

some 𝑣𝑖 , 𝑣𝑗 ∈ 𝑉 , which is a contradiction to the fact that 𝛿

𝐵

𝐴

(𝑣𝑖 ), 𝑇

. . . = ∑𝑣𝑖 ∈𝑉 𝛿𝑇 (𝑣𝑛 , 𝑣𝑖 ),

(𝑣𝑖 ), 𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑣𝑖 , 𝑣𝑗 ∈ 𝑉.

Hence from equations (10) and (11), we have 𝛿

𝐴

(𝑣𝑖 , 𝑣𝑗 ) = min(𝐼

𝑖≠1

(11) 𝐹

𝐵

(𝑣𝑖 , 𝑣𝑗 ) = min(𝑇

By using equation (12)

(𝑣𝑗 ), ∀𝑣𝑖 , 𝑣𝑗 ∈ 𝑉. 𝐹

𝑟

(𝑣𝑖 ) = 𝑒 𝐹

and

𝐵

1 𝑇𝐵 (𝑣2 ,𝑣𝑖 )

} = 𝑚𝑎𝑥{

1 𝑇𝐵 (𝑣3 ,𝑣𝑖 )

} =. . . =

},

𝑇𝐵 (𝑣𝑛 ,𝑣𝑖 )

𝑚𝑎𝑥{

1

} = 𝑚𝑎𝑥{

𝐼𝐵 (𝑣1 ,𝑣𝑖 ) 1

𝑚𝑎𝑥{

},

𝐼𝐵 (𝑣𝑛,𝑣𝑖 )

Now,

15542

1

} = 𝑚𝑎𝑥{

𝐼𝐵 (𝑣2 ,𝑣𝑖 )

1 𝐼𝐵 (𝑣3 ,𝑣𝑖 )

} =. . . =

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 24 (2017) pp. 15536-15543 Š Research India Publications. http://www.ripublication.com đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;{

1

đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;{

1

} = đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;{

1

} = đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;{

1

} =. . . =

[10]

Majumdar P., and Samanta S. K., On similarity and entropy of neutrosophic sets, Journal of Intelligent & Fuzzy Systems, 26(3), (2014) 1245â&#x20AC;&#x201C;1252.

đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;Ľ{đ?&#x203A;żđ?&#x2018;&#x2021; (đ?&#x2018;Ł1 , đ?&#x2018;Łđ?&#x2018;&#x2013; )} = đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;Ľ{đ?&#x203A;żđ?&#x2018;&#x2021; (đ?&#x2018;Ł2 , đ?&#x2018;Łđ?&#x2018;&#x2013; )} = đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;Ľ{đ?&#x203A;żđ?&#x2018;&#x2021; (đ?&#x2018;Ł3 , đ?&#x2018;Łđ?&#x2018;&#x2013; )} = . . . = đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;Ľ{đ?&#x203A;żđ?&#x2018;&#x2021; (đ?&#x2018;Łđ?&#x2018;&#x203A; , đ?&#x2018;Łđ?&#x2018;&#x2013; )}, đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;Ľ{đ?&#x203A;żđ??ź (đ?&#x2018;Ł1 , đ?&#x2018;Łđ?&#x2018;&#x2013; )} = đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;Ľ{đ?&#x203A;żđ??ź (đ?&#x2018;Ł2 , đ?&#x2018;Łđ?&#x2018;&#x2013; )} = đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;Ľ{đ?&#x203A;żđ??ź (đ?&#x2018;Ł3 , đ?&#x2018;Łđ?&#x2018;&#x2013; )} = . . . = đ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;Ľ{đ?&#x203A;żđ??ź (đ?&#x2018;Łđ?&#x2018;&#x203A; , đ?&#x2018;Łđ?&#x2018;&#x2013; )}, đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;{đ?&#x203A;żđ??š (đ?&#x2018;Ł1 , đ?&#x2018;Łđ?&#x2018;&#x2013; )} = đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;{đ?&#x203A;żđ??š (đ?&#x2018;Ł2 , đ?&#x2018;Łđ?&#x2018;&#x2013; )} = đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;{đ?&#x203A;żđ??š (đ?&#x2018;Ł3 , đ?&#x2018;Łđ?&#x2018;&#x2013; )} = . . . = đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;{đ?&#x203A;żđ??š (đ?&#x2018;Łđ?&#x2018;&#x203A; , đ?&#x2018;Łđ?&#x2018;&#x2013; )}. đ?&#x2018;&#x2019;(đ?&#x2018;Ł1 ) = đ?&#x2018;&#x2019;(đ?&#x2018;Ł2 ) = đ?&#x2018;&#x2019;(đ?&#x2018;Ł3 ) =. . . = đ?&#x2018;&#x2019;(đ?&#x2018;Łđ?&#x2018;&#x203A; ).

Therefore G is self-centered.

[11]

Mordeson, J. N. and Nair, P.S., Fuzzy graphs and fuzzy hypergraphs, PhysicaVerlag, Heidelberg, 2001.

[12]

Nagoorgani A., and Radha K., Regular properties of fuzzy graphs, Bulletin of Pure and Applied Sciences,27E(2), (2008) 411 â&#x20AC;&#x201C;419.

[13]

Parvathi ,R and KarunambigaiM.G,Intuitionistic fuzzy Graphs, Journal of Computational Intelligence,Theory and Applications,20, (2006) 139â&#x20AC;&#x201C;150.

CONCLUSION

[14]

Radha K., and Kumaravel N., Some properties of edge regular fuzzy graphs, Jamal Academic ResearchJournal, Special issue, (2014) 121â&#x20AC;&#x201C;127.

[15]

Rosen K. H., Discrete mathematics and its applications, McGraw - Hill, 7th Edition, 2012.

[16]

Wang H., Smarandache F., Zhang Y. Q., and Sunderraman R., Single-valued neutrosophic sets, Multisspace and Multistruct, 4, (2010) 410â&#x20AC;&#x201C;413.

[17]

Smarandache,F., A Unifying field in logics neutrosophy:NeutrosophicProbability,set and logic, Rehoboth:American Research Press,1998.

[18]

Ye J., A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets, Journal of Intelligent & Fuzzy Systems, 26(5), (2014) 2459â&#x20AC;&#x201C;2466.

[19]

Zadeh L. A., Fuzzy sets, Information and control, 8, (1965) 338â&#x20AC;&#x201C;353.

đ??šđ??ľ (đ?&#x2018;Ł1 ,đ?&#x2018;Łđ?&#x2018;&#x2013; )

đ??šđ??ľ (đ?&#x2018;Ł2 ,đ?&#x2018;Łđ?&#x2018;&#x2013; )

đ??šđ??ľ (đ?&#x2018;Ł3 ,đ?&#x2018;Łđ?&#x2018;&#x2013; )

}.

đ??šđ??ľ (đ?&#x2018;Łđ?&#x2018;&#x203A; ,đ?&#x2018;Łđ?&#x2018;&#x2013; )

In this paper, the concepts of length, distance, eccentricity, radius, diameter, status, total status, median and central vertex of a single valued neutrosophic graph have been investigated.

We have presented the concept of self-centered single valued neutrosophic graph. Also some interesting properties of selfcentered single valued neutrosophic graphs followed by some examples.

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15543