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€˛ , đ??ľâ€˛ ) 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?‘&#x; đ?‘‡ (đ??ş) = đ?‘’ đ?‘‡ (đ?‘Łđ?‘– ), ∀đ?‘Łđ?‘– ∈ đ?‘‰.
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?‘&#x; đ??ź (đ??ş) = đ?‘’ đ??ź (đ?‘Łđ?‘– ), ∀đ?‘Łđ?‘– ∈ đ?‘‰. (iii) F- self-centered single valued neutrosophic graph, if every vertex of G is a F- central vertex. (i.e) đ?‘&#x; đ??š (đ??ş) = đ?‘’ đ??š (đ?‘Łđ?‘– ), ∀đ?‘Łđ?‘– ∈ đ?‘‰. (iv) Single valued neutrosophic self-centered graph, if every vertex of G is a central vertex. (i.e) 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?‘– ), ∀đ?‘Łđ?‘– ∈ đ?‘‰. Example 2.2 Consider a single valued neutrosophic graph, đ??ş = (đ??´, đ??ľ) such that đ?‘‰ = {đ?‘Ł1 , đ?‘Ł2 , đ?‘Ł3 , đ?‘Ł4 }, đ??¸ = {(đ?‘Ł1 , đ?‘Ł2 ), (đ?‘Ł2 , đ?‘Ł3 ), (đ?‘Ł3 , đ?‘Ł4 ), (đ?‘Ł4 , đ?‘Ł1 ), (đ?‘Ł1 , đ?‘Ł3 )}.
Path Distance �
(đ?‘‡,đ??ź,đ??š)
Path Distance �
(đ?‘‡,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?‘š[đ?‘ đ?‘‡ (đ??ş)] = đ?‘šđ?‘–đ?‘›{đ?‘ đ?‘‡ (đ?‘˘): đ?‘˘ ∈ đ?‘‰}, (ii) The minimum I-status of G is defined as đ?‘š[đ?‘ đ??ź (đ??ş)] = đ?‘šđ?‘–đ?‘›{đ?‘ đ??ź (đ?‘˘): đ?‘˘ ∈ đ?‘‰}, (iii) The minimum F-status of G is defined as đ?‘š[đ?‘ đ??š (đ??ş)] = đ?‘šđ?‘–đ?‘›{đ?‘ đ??š (đ?‘˘): đ?‘˘ ∈ đ?‘‰}. (iv) The minimum status of G is denoted by đ?‘š[đ?‘ (đ??ş)] and is defined as đ?‘š[đ?‘ (đ??ş)] = (đ?‘š[đ?‘ đ?‘‡ (đ??ş)], đ?‘š[đ?‘ đ??ź (đ??ş)], đ?‘š[đ?‘ 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?‘€[đ?‘ đ?‘‡ (đ??ş)] = đ?‘šđ?‘Žđ?‘Ľ{đ?‘ đ?‘‡ (đ?‘˘): đ?‘˘ ∈ đ?‘‰}, (ii) The maximum I-status of G is defined as đ?‘€[đ?‘ đ??ź (đ??ş)] = đ?‘šđ?‘Žđ?‘Ľ{đ?‘ đ??ź (đ?‘˘): đ?‘˘ ∈ đ?‘‰}, (iii) The maximum F-status of G is defined as đ?‘€[đ?‘ đ??š (đ??ş)] = đ?‘šđ?‘Žđ?‘Ľ{đ?‘ đ??š (đ?‘˘): đ?‘˘ ∈ đ?‘‰}. (iv) The maximum status of G is denoted by đ?‘€[đ?‘ (đ??ş)] and is defined as đ?‘€[đ?‘ (đ??ş)] = (đ?‘€[đ?‘ đ?‘‡ (đ??ş)], đ?‘€[đ?‘ đ??ź (đ??ş)], đ?‘€[đ?‘ 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?‘Ąđ?‘ đ?‘‡ (đ?‘˘) and is defined as đ?‘Ąđ?‘ đ?‘‡ (đ?‘˘) = ∑đ?‘˘âˆˆđ?‘‰ đ?‘ đ?‘‡ (đ?‘˘), The total I-status of a node u of G is denoted by đ?‘Ąđ?‘ đ??ź (đ?‘˘) and is defined as đ?‘Ąđ?‘ đ??ź (đ?‘˘) = ∑đ?‘Łâˆˆđ?‘‰ đ?‘ đ??ź (đ?‘˘), The total F-status of a node u of G is denoted by đ?‘Ąđ?‘ đ??š (đ?‘˘) and is defined as đ?‘Ąđ?‘ đ??š (đ?‘˘) = ∑đ?‘Łâˆˆđ?‘‰ đ?‘ đ??š (đ?‘˘). The total status of G is denoted by đ?‘Ą[đ?‘ (đ??ş)] and is defined as đ?‘Ą[đ?‘ (đ??ş)] = (đ?‘Ąđ?‘ đ?‘‡ (đ?‘˘), đ?‘Ąđ?‘ đ??ź (đ?‘˘), đ?‘Ąđ?‘ đ??š (đ?‘˘)).
đ?‘Ł1 − đ?‘Ł2 đ?‘Ł1 − đ?‘Ł3 đ?‘Ł1 − đ?‘Ł4 (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) (5,5,8) (11,11,12) (6,6,4) đ?‘Ł2 − đ?‘Ł3 đ?‘Ł2 − đ?‘Ł4 đ?‘Ł3 − đ?‘Ł4 (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) (6,6,4) (11,11,12) (5,5,8)
Then the eccentricity of đ?‘Łđ?‘– are đ?‘’(đ?‘Ł1 ) = (11,11,4) , đ?‘’(đ?‘Ł2 ) = (11,11,4), đ?‘’(đ?‘Ł3 ) = (11,11,4), đ?‘’(đ?‘Ł4 ) = (11,11,4). Radius of G is đ?‘&#x;(đ??ş) = (11,11,4) and Diameter of G is đ?‘‘(đ??ş) = (11,11,4). Here đ?‘&#x;(đ??ş) = đ?‘’(đ?‘Łđ?‘– ), ∀đ?‘Łđ?‘– ∈ đ?‘‰ . 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?‘ đ?‘‡ (đ?‘˘) and is defined as đ?‘ đ?‘‡ (đ?‘˘) = ∑đ?‘Łâˆˆđ?‘‰ đ?›żđ?‘‡ (đ?‘˘, đ?‘Ł), (ii) The I-status of a node u of G is denoted by đ?‘ đ??ź (đ?‘˘) and is defined as đ?‘ đ??ź (đ?‘˘) = ∑đ?‘Łâˆˆđ?‘‰ đ?›żđ??ź (đ?‘˘, đ?‘Ł), (iii) The F-status of a node u of G is denoted by đ?‘ đ??š (đ?‘˘) and is defined as đ?‘ đ??š (đ?‘˘) = ∑đ?‘Łâˆˆđ?‘‰ đ?›żđ??š (đ?‘˘, đ?‘Ł), (iv) The status of a node u of G is defined as đ?‘ (đ?‘˘) = (đ?‘ đ?‘‡ (đ?‘˘), đ?‘ đ??ź (đ?‘˘), đ?‘ đ??š (đ?‘˘)).
Definition 2.22 Let đ??ş = (đ??´, đ??ľ) be a connected single valued neutrosophic graph. The median is defined as đ?‘€(đ??ş) = (đ?‘€đ?‘‡ (đ??ş), đ?‘€đ??ź (đ??ş), đ?‘€đ??š (đ??ş)) , where đ?‘€đ?‘‡ (đ??ş) = {đ?‘Łđ?‘– ∈ đ?‘‰: đ?‘šđ?‘–đ?‘›{đ?‘ đ?‘‡ (đ?‘Łđ?‘– )}} , đ?‘€đ??ź (đ??ş) = {đ?‘Łđ?‘– ∈ đ?‘‰: đ?‘šđ?‘–đ?‘›{đ?‘ đ??ź (đ?‘Łđ?‘– )}} , đ?‘€đ??š (đ??ş) = {đ?‘Łđ?‘– ∈ đ?‘‰: đ?‘šđ?‘–đ?‘›{đ?‘ đ??š (đ?‘Łđ?‘– )}}. Example 2.3 Consider a single valued neutrosophic -graph, đ??ş = (đ??´, đ??ľ) such that đ?‘‰ = {đ?‘Ł1 , đ?‘Ł2 , đ?‘Ł3 , đ?‘Ł4 }, đ??¸= {(đ?‘Ł1 , đ?‘Ł2 ), (đ?‘Ł2 , đ?‘Ł3 ), (đ?‘Ł3 , đ?‘Ł1 ), (đ?‘Ł3 , đ?‘Ł4 ), (đ?‘Ł1 , đ?‘Ł4 )}. Here, status of the nodes are đ?‘ (đ?‘Ł1 ) = (22,19,17), đ?‘ (đ?‘Ł2 ) = (27,28,23), đ?‘ (đ?‘Ł3 ) = (26,16,14), đ?‘ (đ?‘Ł4 ) = (29,27,20). The minimum status of G is đ?‘š[đ?‘ (đ??ş)] = (22,16,14) . The maximum status of G is đ?‘€[đ?‘ (đ??ş)] = (29,28,23) . The total status of G is đ?‘Ą[đ?‘ (đ??ş)] = (104,90,74) . The median is đ?‘€(đ??ş) = ({đ?‘Ł1 }, {đ?‘Ł3 }, {đ?‘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?‘Łđ?‘– , đ?‘Łđ?‘–+1 ) is strong single valued neutrosophic for đ?‘– = 1,2, . . . , đ?‘› − 1 and the nodes are alternatively in đ?‘‰1 and đ?‘‰2 , we have đ?‘Łđ?‘› and đ?‘Ł1 ∈ đ?‘‰1 . But this implies that (đ?‘Łđ?‘› , đ?‘Ł1 ) is an edge in đ?‘‰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?‘š[đ?‘ (đ??ş)] = đ?‘€[đ?‘ (đ??ş)]. Example 2.4 Consider a single valued neutrosophic graph, đ??ş = (đ??´, đ??ľ) such that đ?‘‰ = {đ?‘Ł1 , đ?‘Ł2 , đ?‘Ł3 , đ?‘Ł4 }, đ??¸= {(đ?‘Ł1 , đ?‘Ł2 ), (đ?‘Ł2 , đ?‘Ł3 ), (đ?‘Ł3 , đ?‘Ł4 ), (đ?‘Ł4 , đ?‘Ł1 ), (đ?‘Ł1 , đ?‘Ł3 ), (đ?‘Ł2 , đ?‘Ł4 )}. Here, status of the nodes are (đ?‘Ł1 ) = (20,23,15), đ?‘ (đ?‘Ł2 ) = (20,23,15), đ?‘ (đ?‘Ł3 ) = (20,23,15), đ?‘ (đ?‘Ł4 ) = (20,23,15) . The minimum status of G is đ?‘š[đ?‘ (đ??ş)] = (20,23,15) . The maximum status of G is đ?‘€[đ?‘ (đ??ş)] = (20,23,15) . The total status of G is đ?‘Ą[đ?‘ (đ??ş)] = (80,92,60).
đ?‘&#x;(đ??ş) = (
and đ??ź
đ??´
1 �
đ??´
1
,
(đ?‘Ľ) đ??ź
đ??´
1
,
(đ?‘Ľ) đ??š
đ??´
(đ?‘Ľ) are the least value and đ??š
)
đ??´
�
where
(đ?‘Ľ)
đ??´
(đ?‘Ľ)
and
(đ?‘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?‘&#x; where đ?‘‡ đ?‘‡
đ??´
đ??´
�
1
(đ??ş) =
�
đ??´
(đ?‘Łđ?‘– )
,
(đ?‘Łđ?‘– ) is the least. Now fix a vertex đ?‘Łđ?‘– ∈ đ?‘‰ such that
(đ?‘Łđ?‘– ) is least vertex membership value of G.
(1) Consider all the đ?‘Łđ?‘– − đ?‘Łđ?‘— paths P of length n in G, ∀đ?‘Łđ?‘– ∈ đ?‘‰. Case (i) : If đ?‘› = 1 , then đ?‘‡ đ?‘‡
đ??´
đ??ľ
(�� , �� ) = min(�
(đ?‘Łđ?‘– ). Therefore, the T-length of đ?‘ƒ = đ?‘™
�
đ??´
(đ?‘Łđ?‘– , đ?‘Łđ?‘— )) = 1
(đ?‘ƒ) =
�
đ??´
.
(đ?‘Łđ?‘– )
Case (ii) : If đ?‘› > 1, then one of the edges of P possesses the T-strength đ?‘‡ đ??´ (đ?‘Łđ?‘– ) and hence, T-length of a đ?‘Łđ?‘– − đ?‘Łđ?‘— path will exceed
1 �
đ??´
(đ?‘Łđ?‘– )
. That is T-length of P = đ?‘™
�
(đ?‘ƒ) >
1 �
đ??´
(đ?‘Łđ?‘– )
.
Hence �
�
(đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = min(đ?‘™
�
1
(đ?‘ƒ)) =
�
đ??´
(đ?‘Łđ?‘– )
, ∀đ?‘Łđ?‘— ∈ đ?‘‰.
(1)
(2) Let đ?‘Łđ?‘˜ ≠đ?‘Łđ?‘– đ?‘–đ?‘›đ?‘‰. Consider all đ?‘Łđ?‘˜ − đ?‘Łđ?‘— paths Q of length n in G, ∀đ?‘Łđ?‘— ∈ đ?‘‰. Case (i) : If đ?‘› = 1, then đ?‘‡ đ?‘‡ đ?‘™ The median is đ?‘€(đ??ş) = {{đ?‘Ł1 , đ?‘Ł2 , đ?‘Ł3 , đ?‘Ł4 }, {đ?‘Ł1 , đ?‘Ł2 , đ?‘Ł3 , đ?‘Ł4 }, {đ?‘Ł1 , đ?‘Ł2 , đ?‘Ł3 , đ?‘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?‘‰1 and đ?‘‰2 . Suppose that it contains a strong cycle of odd length, say đ?‘Ł1 , đ?‘Ł2 , . . . , đ?‘Łđ?‘› , đ?‘Ł1 for some odd n(vertices). Without loss of generality, let đ?‘Ł1 ∈ đ?‘‰1 . Since
đ??´ đ?‘‡
(�� ), since �
(đ?‘„) =
Case
�
(ii) 1
�
đ??ľ
đ??´
1
(đ?‘Łđ?‘˜+1 ,đ?‘Łđ?‘— )
đ??ľ
(đ?‘Łđ?‘˜ ,đ?‘Łđ?‘— )
(đ?‘Łđ?‘˜ , đ?‘Łđ?‘— ) = min(đ?‘‡
≤
2 �
đ??´
đ??´
(đ?‘Łđ?‘˜ , đ?‘Łđ?‘— )) ≼
(�� ) is the least. Hence T-length of � = 1 �
đ??´
.
(đ?‘Łđ?‘– )
If đ?‘› = 2 ,
: ≤
đ??ľ
(đ?‘Łđ?‘– )
then đ?‘™
, since �
Case (iii) : If đ?‘› > 2, then đ?‘™
�
đ??´
�
(đ?‘„) =
1 �
đ??ľ
(đ?‘Łđ?‘˜ ,đ?‘Łđ?‘˜+1 )
+
(đ?‘Łđ?‘– ) is the least.
(�) ≤
� �
đ??´
(đ?‘Łđ?‘– )
, since �
đ??´
(đ?‘Łđ?‘– ) is
the least. Hence �
�
(đ?‘Łđ?‘˜ , đ?‘Łđ?‘— ) = min(đ?‘™
�
(�)) ≤
1 �
đ??´
(đ?‘Łđ?‘– )
, ∀đ?‘Łđ?‘˜ , đ?‘Łđ?‘— ∈ đ?‘‰.
(2)
, ∀đ?‘Łđ?‘– ∈ đ?‘‰.
(3)
From Equations (1) and (2), we have đ?‘’
15540
�
(�� ) = max(�
�
(đ?‘Łđ?‘– , đ?‘Łđ?‘— )) =
1 �
đ??´
(đ?‘Łđ?‘– )
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?‘&#x; đ?‘‡ (đ??ş) = min(đ?‘’ đ?‘‡ (đ?‘Łđ?‘– )) = đ?‘&#x;
(đ??ş) = đ?‘‡
1 �
đ??´
1 �
đ??´
(đ?‘Łđ?‘– )
, đ?‘¤â„Žđ?‘’đ?‘&#x;đ?‘’đ?‘‡
đ??š
(đ?‘Łđ?‘– ) đ?‘–đ?‘ đ?‘™đ?‘’đ?‘Žđ?‘ đ?‘Ą. đ??´
Next, we claim that G is a I- self-centered single valued neutrosophic graph and đ?‘&#x; đ??ź (đ??ş) =
1 đ??ź
đ??´
(đ?‘Łđ?‘– )
, where đ??ź
the least. Now fix a vertex đ?‘Łđ?‘– ∈ đ?‘‰ such that đ??ź
đ??´
(đ?‘Łđ?‘– ) is
(đ?‘Łđ?‘– ) is least đ??´
vertex membership value of G. (1) Consider all the đ?‘Łđ?‘– − đ?‘Łđ?‘— paths P of length n in G, ∀đ?‘Łđ?‘– ∈ đ?‘‰. Case (i) : If đ?‘› = 1 , then đ??ź đ??ź
đ??´
đ??ľ
(đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = min(đ??ź
đ??´
(đ?‘Łđ?‘– ) . Therefore, the I-length of đ?‘ƒ = đ?‘™ đ??ź (đ?‘ƒ) =
1 đ??ź
đ??´
1 đ??ź
đ??´
(đ?‘Łđ?‘– )
1
. That is I-length of P = đ?‘™ đ??ź (đ?‘ƒ) >
đ??ź
đ??´
(đ?‘Łđ?‘– )
.
1
đ?›ż đ??ź (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = min(đ?‘™ đ??ź (đ?‘ƒ)) =
đ??ź
đ??´
, ∀đ?‘Łđ?‘— ∈ đ?‘‰.
(đ?‘Łđ?‘– )
(đ?‘Łđ?‘– ), since đ??ź đ??´
đ?‘™ đ??ź (đ?‘„) =
đ??ľ
(đ?‘Łđ?‘˜ ,đ?‘Łđ?‘—)
≤
1 đ??ź
đ??´
.
đ??ź
đ??´
, since đ??ź
1 đ??ź
đ??ľ
(đ?‘Łđ?‘˜ ,đ?‘Łđ?‘˜+1 )
+
1 đ??ź
đ??ľ
(đ?‘Łđ?‘˜+1 ,đ?‘Łđ?‘— )
≤
(đ?‘Łđ?‘– ) is the least. đ??´
(đ?‘Łđ?‘– )
Case (iii): If đ?‘› > 2, then đ?‘™ đ??ź (đ?‘„) ≤
đ??´
(đ?‘Łđ?‘– )
đ??š
đ?‘› đ??ź
đ??´
(đ?‘Łđ?‘– )
, since đ??ź
đ??´
(đ?‘Łđ?‘– ) is
1 đ??ź
đ?‘’ đ??ź (đ?‘Łđ?‘– ) = max(đ?›ż đ??ź (đ?‘Łđ?‘– , đ?‘Łđ?‘— )) =
đ??´
(đ?‘Łđ?‘– )
, ∀đ?‘Łđ?‘˜ , đ?‘Łđ?‘— ∈ đ?‘‰.
đ?‘&#x; đ??ź (đ??ş) =
1 đ??´
(đ?‘Łđ?‘– )
đ??š
đ??´
(đ?‘Łđ?‘– ), since đ??š
đ?‘„=đ?‘™
đ??š
(đ?‘„) =
(ii):
1 đ??ź
đ??´
(đ?‘Łđ?‘– )
1 , đ?‘¤â„Žđ?‘’đ?‘&#x;đ?‘’đ??ź đ??ź đ??´ (đ?‘Łđ?‘– )
đ??´
(đ?‘Łđ?‘– )
.
đ??š
(đ?‘ƒ) >
1 đ??š
đ??´
(đ?‘Łđ?‘– )
.
đ??š
đ??´
(đ?‘Łđ?‘– )
, ∀đ?‘Łđ?‘— ∈ đ?‘‰.
(7)
đ??ľ
đ??´
đ??š
đ??ľ
≼
1 đ??š
đ??´
đ?‘›=2 , đ??´
(đ?‘Łđ?‘˜ , đ?‘Łđ?‘— ) = max(đ??š
đ??´
(đ?‘Łđ?‘˜ , đ?‘Łđ?‘— )) ≤
(đ?‘Łđ?‘– ) is the greatest. Hence F-length of
(đ?‘Łđ?‘˜ ,đ?‘Łđ?‘— )
2 đ??š
đ??ľ
. đ?‘™
then
, since đ??š
(đ?‘Łđ?‘– )
Case (iii): If đ?‘› > 2, then đ?‘™
(đ?‘Łđ?‘– )
đ??š
đ??´
đ??š
(đ?‘„) =
1 đ??š
đ??ľ
(đ?‘Łđ?‘˜ ,đ?‘Łđ?‘˜+1 )
+
(đ?‘Łđ?‘– ) is the greatest. đ?‘›
(�) ≼
đ??š
đ??´
, since đ??š
(đ?‘Łđ?‘– )
đ??´
(đ?‘Łđ?‘– ) is
the greatest. Hence �
đ??š
(đ?‘Łđ?‘˜ , đ?‘Łđ?‘— ) = min(đ?‘™
đ??š
1
(�)) ≼
đ??š
đ??´
(đ?‘Łđ?‘– )
, ∀đ?‘Łđ?‘˜ , đ?‘Łđ?‘— ∈ đ?‘‰.
(8)
đ?‘’
đ??š
(�� ) = min(�
, ∀đ?‘Łđ?‘– ∈ đ?‘‰.
(đ??ş) =
(6)
đ??´
(đ?‘Łđ?‘– )
, where đ??š
đ??´
(đ?‘Łđ?‘– , đ?‘Łđ?‘— )) =
1 đ??š
đ??´
(đ?‘Łđ?‘– )
, ∀đ?‘Łđ?‘– ∈ đ?‘‰.
(9)
Hence G is a F-self-centered single valued neutrosophic graph. Now, đ?‘&#x; đ??š (đ??ş) = min(đ?‘’ đ??š (đ?‘Łđ?‘– ))
đ??š
(đ??ş) =
1 đ??š
đ??´
(đ?‘Łđ?‘– )
1 đ??š
đ??´
(đ?‘Łđ?‘– )
, đ?‘ đ?‘–đ?‘›đ?‘?đ?‘’đ?‘?đ?‘Śđ?‘’đ?‘žđ?‘˘đ?‘Žđ?‘Ąđ?‘–đ?‘œđ?‘› (9) , đ?‘¤â„Žđ?‘’đ?‘&#x;đ?‘’đ??š
đ??´
(đ?‘Łđ?‘– ) đ?‘–đ?‘ đ?‘”đ?‘&#x;đ?‘’đ?‘Žđ?‘Ąđ?‘’đ?‘ đ?‘Ą.
Theorem 2.3 A single valued neutrosophic graph đ??ş = (đ??´, đ??ľ) is a self-centered single valued neutrosophic graph iff đ?›ż đ?‘‡ (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) ≤ đ?‘&#x; đ?‘‡ (đ??ş) , đ?›ż đ??ź (đ?‘Łđ?‘– , đ?‘Łđ?‘— ) ≤ đ?‘&#x; đ??ź (đ??ş) and đ?›ż
(đ?‘Łđ?‘– ) đ?‘–đ?‘ đ?‘™đ?‘’đ?‘Žđ?‘ đ?‘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?‘ƒ)) =
1
If ≼
(đ?‘Łđ?‘˜+1 ,đ?‘Łđ?‘—)
đ?‘&#x;
, đ?‘ đ?‘–đ?‘›đ?‘?đ?‘’đ?‘?đ?‘Śđ?‘’đ?‘žđ?‘˘đ?‘Žđ?‘Ąđ?‘–đ?‘œđ?‘› (6)
neutrosophic graph and đ?‘&#x;
đ??š
Case (i): If đ?‘› = 1, then đ??š
(5)
Hence G is a I-self-centered single valued neutrosophic graph. Now, đ?‘&#x; đ??ź (đ??ş) = min(đ?‘’ đ??ź (đ?‘Łđ?‘– )) đ??ź
1 đ??š
(2) Let đ?‘Łđ?‘˜ ≠đ?‘Łđ?‘– đ?‘–đ?‘›đ?‘‰. Consider all đ?‘Łđ?‘˜ − đ?‘Łđ?‘— paths Q of length n
=
From Equations (4) and (5), we have
=
. That is F-length of P= đ?‘™
(đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = min(đ?‘™
the least. Hence đ?›ż đ??ź (đ?‘Łđ?‘˜ , đ?‘Łđ?‘— ) = min(đ?‘™ đ??ź (đ?‘„)) ≤
(đ?‘ƒ) =
From Equations (7) and (8), we have
(đ?‘Łđ?‘– )
Case (ii): If đ?‘› = 2, then đ?‘™ đ??ź (đ?‘„) = 2
(đ?‘Łđ?‘˜ , đ?‘Łđ?‘— )) ≼
đ??´
đ??ź
1
(đ?‘Łđ?‘– ) is the least. Hence I-length of đ?‘„ = đ??´
1 đ??ź
(đ?‘Łđ?‘˜ , đ?‘Łđ?‘— ) = min(đ??ź
đ??š
(đ?‘Łđ?‘– , đ?‘Łđ?‘— )) =
đ??´
Hence
đ??š
in G, ∀đ?‘Łđ?‘— ∈ đ?‘‰. đ??ľ
1
exceed
(4)
(2) Let đ?‘Łđ?‘˜ ≠đ?‘Łđ?‘– đ?‘–đ?‘›đ?‘‰. Consider all đ?‘Łđ?‘˜ − đ?‘Łđ?‘— paths Q of length n Case (i) : If đ?‘› = 1 , then đ??ź
(đ?‘Łđ?‘– , đ?‘Łđ?‘— ) = max(đ??š
(đ?‘Łđ?‘– ). Therefore, the F - length of đ?‘ƒ = đ?‘™
Case
Hence
đ??ź
đ??´
đ??ľ
in G, ∀đ?‘Łđ?‘— ∈ đ?‘‰.
.
(đ?‘Łđ?‘– )
Case (ii) : If đ?‘› > 1, then one of the edges of P possesses the Istrength đ??ź đ??´ (đ?‘Łđ?‘– ) and hence, I-length of a đ?‘Łđ?‘– − đ?‘Łđ?‘— path will exceed
(đ?‘Łđ?‘– ) is
Case (ii): If đ?‘› > 1, then one of the edges of P possesses the Fstrength đ??ź đ??´ (đ?‘Łđ?‘– ) and hence, F-length of a đ?‘Łđ?‘– − đ?‘Łđ?‘— path will
�
(đ?‘Łđ?‘– , đ?‘Łđ?‘— )) =
đ??´
greatest vertex membership value of G. (1) Consider all the đ?‘Łđ?‘– − đ?‘Łđ?‘— paths P of length n in G, ∀đ?‘Łđ?‘– ∈ đ?‘‰. Case (i): If đ?‘› = 1 , then đ??š
, đ?‘ đ?‘–đ?‘›đ?‘?đ?‘’đ?‘?đ?‘Śđ?‘’đ?‘žđ?‘˘đ?‘Žđ?‘Ąđ?‘–đ?‘œđ?‘›(3)
(đ?‘Łđ?‘– )
the greatest. Now fix a vertex đ?‘Łđ?‘– ∈ đ?‘‰ such that đ??š
(đ?‘Łđ?‘– ) is
đ??š
(đ?‘Łđ?‘– , đ?‘Łđ?‘— ) ≼ đ?‘&#x;
đ??š
(đ??ş)∀đ?‘Łđ?‘– , đ?‘Łđ?‘— ∈ đ?‘‰.
Proof.⇒ We assume that G is self-centered single valued neutrosophic graph G. That is đ?‘’ đ?‘‡ (đ?‘Łđ?‘– ) = đ?‘’ đ?‘‡ (đ?‘Łđ?‘— ), đ?‘’ đ??ź (đ?‘Łđ?‘– ) = đ?‘’ đ??ź (đ?‘Łđ?‘— ), đ?‘’ đ?‘’
�
(đ?‘Łđ?‘– ) = đ?‘’
đ??š
(đ?‘Łđ?‘— ), ∀đ?‘Łđ?‘– , đ?‘Łđ?‘— ∈ đ?‘‰, đ?‘&#x;
(đ?‘Łđ?‘– ), đ?‘&#x; đ??ź (đ??ş) = đ?‘’ đ??ź (đ?‘Łđ?‘– ) and đ?‘&#x;
Now
15541
đ??š
we
wish
to
đ??š
show
�
(đ??ş) = đ?‘’ that
(đ??ş) = đ??š
(đ?‘Łđ?‘– ), ∀đ?‘Łđ?‘– ∈ đ?‘‰ . đ?›ż
�
(�� , �� ) ≤
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 ���{
1
đ?‘šđ?‘–đ?‘›{
1
} = đ?‘šđ?‘–đ?‘›{
1
} = đ?‘šđ?‘–đ?‘›{
1
} =. . . =
[10]
Majumdar P., and Samanta S. K., On similarity and entropy of neutrosophic sets, Journal of Intelligent & Fuzzy Systems, 26(3), (2014) 1245–1252.
đ?‘šđ?‘Žđ?‘Ľ{đ?›żđ?‘‡ (đ?‘Ł1 , đ?‘Łđ?‘– )} = đ?‘šđ?‘Žđ?‘Ľ{đ?›żđ?‘‡ (đ?‘Ł2 , đ?‘Łđ?‘– )} = đ?‘šđ?‘Žđ?‘Ľ{đ?›żđ?‘‡ (đ?‘Ł3 , đ?‘Łđ?‘– )} = . . . = đ?‘šđ?‘Žđ?‘Ľ{đ?›żđ?‘‡ (đ?‘Łđ?‘› , đ?‘Łđ?‘– )}, đ?‘šđ?‘Žđ?‘Ľ{đ?›żđ??ź (đ?‘Ł1 , đ?‘Łđ?‘– )} = đ?‘šđ?‘Žđ?‘Ľ{đ?›żđ??ź (đ?‘Ł2 , đ?‘Łđ?‘– )} = đ?‘šđ?‘Žđ?‘Ľ{đ?›żđ??ź (đ?‘Ł3 , đ?‘Łđ?‘– )} = . . . = đ?‘šđ?‘Žđ?‘Ľ{đ?›żđ??ź (đ?‘Łđ?‘› , đ?‘Łđ?‘– )}, đ?‘šđ?‘–đ?‘›{đ?›żđ??š (đ?‘Ł1 , đ?‘Łđ?‘– )} = đ?‘šđ?‘–đ?‘›{đ?›żđ??š (đ?‘Ł2 , đ?‘Łđ?‘– )} = đ?‘šđ?‘–đ?‘›{đ?›żđ??š (đ?‘Ł3 , đ?‘Łđ?‘– )} = . . . = đ?‘šđ?‘–đ?‘›{đ?›żđ??š (đ?‘Łđ?‘› , đ?‘Łđ?‘– )}. đ?‘’(đ?‘Ł1 ) = đ?‘’(đ?‘Ł2 ) = đ?‘’(đ?‘Ł3 ) =. . . = đ?‘’(đ?‘Łđ?‘› ). Therefore G is self-centered.
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CONCLUSION
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đ??šđ??ľ (đ?‘Ł1 ,đ?‘Łđ?‘– )
đ??šđ??ľ (đ?‘Ł2 ,đ?‘Łđ?‘– )
đ??šđ??ľ (đ?‘Ł3 ,đ?‘Łđ?‘– )
}.
đ??šđ??ľ (đ?‘Łđ?‘› ,đ?‘Łđ?‘– )
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. REFERENCES [1]
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