PERFECT AND STATUS IN SINGLE VALUED NEUTROSOPHIC GRAPHS

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1 International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue 1, January 2018, pp , Article ID: IJMET_09_01_120 Available online at ISSN Print: and ISSN Online: IAEME Publication Scopus Indexed PERFECT AND STATUS IN SINGLE VALUED NEUTROSOPHIC GRAPHS V Krishnaraj Assistant Professor, Department of Mathematics, Sona College of Technology, Salem, Tamil Nadu, India RVikramaprasad Assistant Professor, Department of Mathematics, Government Arts College, Salem, Tamil Nadu, India R Dhavaseelan Assistant Professor, Department of Mathematics, Sona College of Technology, Salem, Tamil Nadu, India ABSTRACT In this paper, we introduce the concepts of perfect single valued neutrosophic, complete perfect single valued neutrosophic vertices, edges and graphs We investigated some properties of status in complete perfect single valued neutrosophic graphs Key words: Perfect SVN vertex, perfect SVN edge, complete perfect SVN graphs, status in SVN graphs AMS Classification: 05C07, 68R10, 03E72 Cite this Article: V Krishnaraj, R Vikramaprasad and R Dhavaseelan, Perfect and Status in Single Valued Neutrosophic Graphs, International Journal of Mechanical Engineering and Technology 9(1), 2018, pp INTRODUCTION The fundamental concept of a fuzzy relation was defined by zadeh [14] Atanassov K introduced the concepts of Intuitionistic fuzzy sets and systems [5] Parvathi R, Karunambigai MG introduced the idea of a intuitionistic fuzzy relations and intuitionistic fuzzy graphs(ifgs) [12] Smarandache introduced the concept of neutrosophic sets [13] The concepts of certain types of neutrosophic graphs was introduced by R Dhavaseelan, et al [8] A Nagoor Gani and M Basheer Ahmed [10] defined perfect fuzzy graph and complete perfect fuzzy graph and its properties VKrishnaraj, et al [9] defined status, self-center and various properties in single valued neutrosophic graphs In this paper, we introduce the concepts of and -perfect single valued neutrosophic vertices, and -perfect single valued neutrosophic edges, editor@iaemecom

2 Perfect and Status in Single Valued Neutrosophic Graphs perfect single valued neutrosophic graphs, -perfect single valued neutrosophic graphs, complete and -perfect single valued neutrosophic graphs, complete and -perfect single valued neutrosophic graphs, and -perfect single valued neutrosophic graphs, complete and -perfect single valued neutrosophic graphs, complete perfect single valued neutrosophic graphs and its properties 2 PRELIMINARIES Definition 21 [1] A Single Valued Neutrosophic (SVN) Graph with underlying set V is defined to be a pair G = (A, B) where 1 The function denote the degree of truthmembership, degree of indeterminacy-membership and falsity-membership of the element, respectively, and 2 The functions are defined by and Definition 22 [3] A single valued neutrosophic graph G= (A, B) is called complete single valued neutrosophic graph if and Definition 23 [9] A connected SVN graph G is a T- self centered SVN graph, if every vertex of G is a T- central vertex (ie) I- self centered SVN graph, if every vertex of G is a I- central vertex (ie) F- self centered SVN graph, if every vertex of G is a F- central vertex (ie) is called self centered SVN graph, if every vertex of G is a central vertex (ie) Definition 24 [9] Let be a connected SVN graph The T-status of a vertex u of G is denoted by and is defined as, The I-status of a vertex u of G is denoted by and is defined as, The F-status of a vertex u of G is denoted by and is defined as, The status of a vertex u of G is defined as Definition 25 [9] Let be a connected SVN graph The minimum T-status of G is defined as The minimum I-status of G is defined as The minimum F-status of G is defined as The minimum status of G is denoted by and is defined as Definition 26 [9] Let be a connected SVN graph The maximum T-status of G is defined as The maximum I-status of G is defined as The maximum F-status of G is defined as The maximum status of G is denoted by and is defined as Definition 27 [9] Let be a connected SVN graph The total T-status of a vertex u of G is denoted by and is defined as, The total I-status of a editor@iaemecom

3 V Krishnaraj, R Vikramaprasad and R Dhavaseelan vertex u of G is denoted by and is defined as, The total F-status of a vertex u of G is denoted by and is defined as The total status of G is denoted by and is defined as Definition 28 [9] Let be a connected SVN graph The median is defined as, where,, Definition 29 [9] A connected SVN graph is a self-median if all the vertices have the same status In other words, a connected SVN-graph is self-median if and only if 3 PERFECT AND COMPLETE PERFECT IN SINGLE VALUED NEUTROSOPHIC GRAPHS Definition 31 Let be a connected SVN graph A vertex is called a (1) -perfect SVN vertex if, (2) -perfect SVN vertex if, (3) -perfect SVN vertex if for some An edge is called a (1) -perfect SVN edge if, (2) -perfect SVN edge if, (3) -perfect SVN edge if for some Example 31 Consider a SVN-graph, such that Definition 32 A connected SVN graph is called a -perfect SVN graph if, -perfect SVN graph if, -perfect SVN graph if Example 32 Consider SVN-graphs editor@iaemecom

4 Perfect and Status in Single Valued Neutrosophic Graphs T A perfect SVN graph I A perfect SVN graph F A perfect SVN graph Definition 33 A connected SVN graph is called a -perfect SVN graph if, -perfect SVN graph if, -perfect SVN graph if Example 33 Consider SVN graphs, such that Remark 31 (i) Every -perfect SVN graph is -perfect SVN graph, Converse need not be true (ii) Every -perfect SVN graph is -perfect SVN graph, Converse need not be true (iii) Every -perfect SVN graph is -perfect SVN graph, Converse need not be true Definition 34 A connected SVN-graph is called a (1) Complete -perfect SVN graph if, (2) Complete -perfect SVN graph if, (3) Complete -perfect SVN graph if Example 34 Consider SVN-graphs, such that editor@iaemecom

5 V Krishnaraj, R Vikramaprasad and R Dhavaseelan, such that, Complete T B perfect SVNG Complete T B perfect SVNG Complete T B perfect Definition 35 A connected SVN-graph is called a -perfect single valued neutrosophic graph if it has -perfect single valued neutrosophic graph and -perfect single valued neutrosophic graph Definition 36 A connected SVN-graph is called a -perfect single valued neutrosophic graph if it has -perfect single valued neutrosophic graph and -perfect single valued neutrosophic graph Definition 37 A connected SVN-graph is called a -perfect single valued neutrosophic graph if it has -perfect single valued neutrosophic graph and -perfect single valued neutrosophic graph Example 35 Consider SVN-graphs Definition 38 A connected SVN-graph is called a perfect single valued neutrosophic graph if it has, and -perfect single valued neutrosophic graph Example 36 Consider SVN-graphs, such that editor@iaemecom

6 Perfect and Status in Single Valued Neutrosophic Graphs Remark 32 A connected SVN-graph is called a complete -perfect single valued neutrosophic graph if it has complete -perfect single valued neutrosophic graph Remark 33 A connected SVN-graph is called a complete -perfect single valued neutrosophic graph if it has complete -perfect single valued neutrosophic graph Remark 34 A connected SVN-graph is called a complete -perfect single valued neutrosophic graph if it has -perfect single valued neutrosophic graph and complete - perfect single valued neutrosophic graph Example 37 Consider SVN-graphs, such that, Definition 39 A connected SVN-graph is called a complete perfect single valued neutrosophic graph if it has complete, and -perfect single valued neutrosophic graph Example 38 Consider a SVN-graph 4 STATUS IN SINGLE VALUED NEUTROSOPHIC GRAPHS Theorem 41 If Proof Let is a complete single valued neutrosophic graph, then, be a complete SVN graph, where Since is complete SVN graph, where and are the least value and is greatest value editor@iaemecom

7 V Krishnaraj, R Vikramaprasad and R Dhavaseelan Hence Example 41 Consider a SVN-graph Eccentricity of are,,,, Radius of G is, Status of are,,,, Maximum status of G is ( ) and Minimum status of G is ( ) Therefore Theorem 42 Every complete perfect single valued neutrosophic graph is a self-centered single valued neutrosophic graph Proof Since G be a complete perfect single valued neutrosophic graph, Therefore and Hence G is self centered single valued neutrosophic graph Theorem 43 Every complete perfect single valued neutrosophic graph is a self median single valued neutrosophic graph Proof Since G be a complete perfect single valued neutrosophic graph, Therefore Status of are Therefore Hence G is a self median single valued neutrosophic graph editor@iaemecom

8 Perfect and Status in Single Valued Neutrosophic Graphs Theorem 44 The -radius of a complete -perfect SVNG with n vertices is always one Proof Since is a complete -perfect SVNG with n vertices, Therefore, Hence -radius of a complete -perfect SVNG with n vertices is always one Corollary 41 The -radius of a complete -perfect SVNG with n vertices is always one Corollary 42 The -radius of a complete -perfect SVNG with n vertices is always one Theorem 45 The -diameter of a complete -perfect SVNG with n vertices is always one Proof Since is a complete -perfect SVNG with n vertices, Therefore, Hence -diameter of a complete -perfect SVNG with n vertices is always one Corollary 43 The -diameter of a complete -perfect SVNG with n vertices is always one Corollary 44 The -diameter of a complete -perfect SVNG with n vertices is always one Theorem 46 The -status of any vertex in a complete -perfect SVNG is Proof Let be a complete -perfect SVNG with n vertices Distance Therefore -status of any vertex of G is Hence the -status of any vertex in a complete -perfect SVNG is Corollary 45 The -status of any vertex in a complete -perfect SVNG is Corollary 46 The -status of any vertex in a complete -perfect SVNG is Theorem 47 The total -status of any vertex in a complete -perfect SVNG is Proof Let be a complete -perfect SVNG with n vertices Therefore, editor@iaemecom

9 V Krishnaraj, R Vikramaprasad and R Dhavaseelan Hence the total -status of any vertex in a complete -perfect SVNG is Corollary 47 The total -status of any vertex in a complete -perfect SVNG is Corollary 48 The total -status of any vertex in a complete -perfect SVNG is 5 CONCLUSIONS In this paper, concepts of perfect single valued neutrosophic, complete perfect single valued neutrosophic vertices, edges and graphs have been introduced Also investigated some properties of status in complete perfect single valued neutrosophic graphs followed by some examples REFERENCES [1] M Akram and G Shahzadi, Operations on single-valued neutrosophic graphs, Journal of Uncertain System, 11(2), (2017) 1-26 [2] M Akram and S Shahzadi, Neutrosophic soft graphs with application, Journal of Intelligent & Fuzzy Systems, 32(1), (2017) [3] M Akram, Single-valued neutrosophic planar graphs, International Journal of Algebra and Statistics, 5(2), (2016) [4] M Akram and S Shahzadi, Representation of graphs using intuitionistic neutrosophic soft sets, Journal of Mathematical Analysis, 7(6), (2016) [5] Atanassov K, Intuitionistic fuzzy sets, Fuzzy sets and Systems, 20, (1986) [6] Broumi, S, Talea, M, A Bakali and F Smarandache, Single Valued Neutrosophic Graphs: Degree, Order and Size, IEEE International Conference on Fuzzy Systems, (2016) [7] Atanassov K, Intuitionistic fuzzy sets, Fuzzy sets and Systems, 20, (1986) [8] Dhavaseelan, R, Vikramaprasad, R, and V Krishnaraj, Certain types of neutrosophic graphs, International Journal of Mathematical Sciences and Applications, 5(2), (2015) [9] VKrishnaraj, RVikramaprasad and RDhavaseelan, Self-Centered Single Valued Neutrosophic Graphs, International Journal of Applied Engineering Research, 12(24), (2017) [10] Nagoor Gani A, and Basheer Ahamed M, Perfect Fuzzy Graphs, Bulletin of Pure and Applied Sciences 28(1), (2009) [11] Nagoor Gani A and Shajitha Begum S, Perfect Intuitionistic Fuzzy Graphs, Bulletin of Pure and Applied Sciences 30(2), (2011) [12] Parvathi,R and Karunambigai MG, Intuitionistic fuzzy Graphs, Journal of Computational Intelligence,Theory and Applications,20, (2006) [13] F Smarandache, Neutrosophy and Neutrosophic Logic, First International Conference on Neutrosophy, Neutrosophic Logic, Set, Probability and Statistics University of New Mexico, Gallup, NM 87301, USA (2002) [14] Zadeh L A, Fuzzy sets, Information and control, 8, (1965), editor@iaemecom

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