Certain Notions of Energy in Single-Valued Neutrosophic Graphs

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1 axioms Article Certai Notios of Eergy i Sigle-Valued Neutrosophic Graphs Sumera Naz 1 Muhammad Akram 1 * ad Floreti Smaradache 2 1 Departmet of Mathematics Uiversity of the Pujab New Campus Lahore Pakista; sumeraaz@gmailcom 2 Mathematics & Sciece Departmet Uiversity of New Mexico 05 Gurley Ave Gallup NM 8301 USA; fsmaradache@gmailcom * Correspodece: makram@pucitedupk; Tel: Received: 12 Jue 2018; Accepted: 16 July 2018; Published: 20 July 2018 Abstract: A sigle-valued eutrosophic set is a istace of a eutrosophic set which provides us a additioal possibility to represet ucertaity imprecise icomplete ad icosistet iformatio existig i real situatios I this research study we preset cocepts of eergy Laplacia eergy ad sigless Laplacia eergy i sigle-valued eutrosophic graphs SVNGs) describe some of their properties ad develop relatioship amog them We also cosider practical examples to illustrate the applicability of the our proposed cocepts Keywords: sigle-valued eutrosophic graph; eergy; Laplacia eergy; sigless Laplacia eergy; decisio-makig 1 Itroductio Smaradache [1] origially itroduced the otio of eutrosophic set NS) from philosophical poit of view Wag et al [2] put forth the otio of a sigle-valued eutrosophic set SVNS) from a scietific or egieerig poit of view as a subclass of the NS ad a extesio of ituitioistic fuzzy set IFS) [3] ad provide its various properties The promiet characteristic of SVNS is that a truth-membership a idetermiacy-membership ad a falsity-membership degree i [01] are idepedetly assiged to each elemet i the set IFSs caot deal with all types of ucertaity such as idetermiate ad icosistet iformatio existig commoly i real situatios For istace if durig a votig process there are sixtee voters seve vote aye six vote blackball ad three are udecided Accordig to sigle-valued eutrosophic otatio it ca be represeted as u This iformatio is beyod the scope of IFS That is why the cocept of SVNS is more extesive tha IFS NS particularly SVNS has attracted sigificat iterest from researchers i recet years It has bee widely applied i various fields icludig iformatio fusio i which data are combied from differet sesors [4] cotrol theory [5] image processig [6] medical diagosis [] decisio makig [8] ad graph theory [910] A graph is a mathematical object cotaiig poits vertices) ad coectios edges) For istace the vertices could represet commuicatio ceters with edges depictig commuicatio liks Graph spectra is oe of the most importat cocepts of graph theory Gutma [11] itroduced the otio of eergy of a graph i chemistry because of its relevace to the total π-electro eergy of certai molecules ad foud upper ad lower bouds for the eergy of graphs [12] Later Gutma ad Zhou [13] defied the Laplacia eergy of a graph as the sum of the absolute values of the differeces of average vertex degree of G to the Laplacia eigevalues of G Sigless Laplacia eergy of a graph was defied i [14] However i may real-life applicatios there is a variety of o-determiistic iformatio due to the icrease of system complexity Sometimes the coectio betwee two objects caot be fully determied ad to verify the properties of the graph traditioal Axioms ; doi:103390/axioms wwwmdpicom/joural/axioms

2 Axioms of 30 methods are useless Erdös [15] used the probability theory to deal with this problem Meawhile after the iceptio of fuzzy sets by Zadeh [16] the cocept of fuzzy graph was put forth by Kaufma [1] ad Rosefeld [18] to hadle the fuzzy pheomea i graphs Fuzzy graphs are useful i represetig structures of relatioships betwee objects where the existece of a cocrete object ad relatioship betwee two objects are ucertai or obscure Ajali ad Mathew [19] ivestigated the eergy of a graph withi the framework of fuzzy set theory Laplacia eergy of a fuzzy graph was defied by Sharbaf ad Fayazi [20] Later o may geeralized fuzzy graphs [21 31] have bee itroduced i literature Amog these extesios the otio of ituitioistic fuzzy graph IFG) whose vertex set ad edge set specify a degree of membership a degree of o-membership ad a degree of hesitacy was proposed by Parvathi ad Karuambigai [21] ad Akram ad Davvaz [23] Praba et al [32] defied the eergy of IFGs Basha ad Kartheek [33] geeralized the cocept of the Laplacia eergy of fuzzy graph to the Laplacia eergy of a IFG Whe descriptio of the objects or their relatios or both is idetermiate ad icosistet it caot be hadled by fuzzy graphs ad IFGs To overcome this shortcomig of the IFGs Akram et al [24] exteded the cocept of IFGs to SVNGs ad put forward may ew cocepts related to SVNGs ad its extesios [24 26] Naz et al [10] itroduced the cocept of sigle-valued eutrosophic digraphs SVNDGs) I this research study we preset cocepts of eergy Laplacia eergy ad sigless Laplacia eergy i sigle-valued eutrosophic graphs SVNGs) describe some of their properties ad develop relatioship amog them We also cosider practical examples to illustrate the applicability of the our proposed cocepts Defiitio 1 [2] Let Z be a space of poits objects) with a geeric elemet i Z deoted by u A SVNS X i Z is characterized by a truth-membership fuctio T X a idetermiacy-membership fuctio I X ad a falsity-membership fuctio F X For each poit u X T X u) I X u) F X u) [0 1] Therefore a SVNS X i Z ca be writte as X = { u T X u) I X u) F X u) u Z} Defiitio 2 [24] A SVNG o a o-empty set Z is a pair G = X Y) where X is a sigle-valued eutrosophic set SVNS) i Z ad Y is a sigle-valued eutrosophic relatio o Z such that T Y uv) mi{t X u) T X v)} I Y uv) max{i X u) I X v)} F Y uv) max{f X u) F X v)} for all u v Z X ad Y are called the sigle-valued eutrosophic vertex set ad the sigle-valued eutrosophic edge set of G respectively Here Y is a symmetric sigle-valued eutrosophic relatio o X If Y is ot symmetric o X the D = X Y ) is called SVNDG We have used stadard defiitios ad termiologies i this paper For more details ad backgroud the readers are referred to [34 40] 2 Eergy of Sigle-Valued Neutrosophic Graphs I this sectio we defie ad ivestigate the eergy of a graph withi the framework of SVNS theory ad discuss its properties Defiitio 3 The adjacecy matrix AG) of a SVNG G = X Y) is defied as a square matrix AG) = [a jk ] a jk = T Y u j u k ) I Y u j u k ) F Y u j u k ) where T Y u j u k ) I Y u j u k ) ad F Y u j u k ) represet the stregth of relatioship stregth of udecided relatioship ad stregth of o-relatioship betwee u j ad u k respectively

3 Defiitio 21 The adjacecy matrix AG) of a SVNG G = X Y ) is defied as a square matrix AG) Axioms = [a jk 2018 ] a jk 50= T Y u j u k ) I Y u j u k ) F Y u j u k ) where T Y u j u k ) I Y u j u k ) ad F Y u j u k ) 3represet of 30 the stregth of relatioship stregth of udecided relatioship ad stregth of o-relatioship betwee u j ad u k respectively The adjacecy matrix of a SVNG ca be expressed as three matrices first matrix cotais the etries as The truth-membership adjacecy matrix values of asecod SVNGcotais ca be expressed the etries as three idetermiacy-membership matrices first matrixvalues cotais adthe theetries third as truth-membership cotais the etries values as falsity-membership secod cotais the values etries ie as AG) idetermiacy-membership = AT Y u j u k )) AI Y u j uvalues k )) AFad Y u j uthe k )) third cotais the etries as falsity-membership values ie AG) = AT Y u j u k )) AI Y u j u k )) AF Y u j u k )) Defiitio Defiitio 22 4 The The spectrum spectrum of of adjacecy adjacecy matrix matrix of a SVNG of a SVNG AG) is AG) defied is defied as M N as O M where N O M N where ad M N ad O are O are the sets theof sets eigevalues of eigevalues of AT Y of u j AT u k )) AI Y u j u k )) ad AF Y u j u k )) respectively Y u j u k )) AI Y u j u k )) ad AF Y u j u k )) respectively Example 21 Cosider aa graph G G = V = V E) E) where where V = V {u= 1 u{u 2 1 u 3 u 24 u 35 u 46 u 5 } uad 6 ue } = ad {u 1 ue 2 = u 2 u{u 3 1 u 32 u 4 u 1 u 3 u 1 u 4 u 4 u 1 u 1 5 u 1 u 15 u u 6 1 u 62 u 31 u 3 u 34 u u 5 4 u 35 u 6 u u 5 u 3 6 } u 2 Let u 5 ug 5 u= 6 X u 6 u Y u) 4 ube } alet SVNG G = X o Y) V be asa show SVNG o i V Fig as show 1 defied by i Figure 1 defied by X u 1 u 2 u 3 u 4 u 5 u 6 u T X X 06u 1 04 u 2 05 u 3 06 u 4 u03 5 u 6 02 u 02 I X T X F X I X F X Y u 1 u 2 u 2 u 3 u 3 u 4 u 4 u 1 u 1 u 5 u 1 u 6 u 1 u u 3 u 5 u 3 u 6 u 3 u u 2 u 5 u 5 u 6 u 6 u u 4 u T Y Y02 u 1 u 2 03u 2 u 3 03 u 3 u 4 05 u 4 u 1 u02 1 u 5 u 1 01 u 6 u 1 u02 u 3 u02 5 u 3 u 6 01u 3 u 02 u 2 u 5 02 u 5 u 6 u02 6 u u 4 01 u 02 I Y T01 Y F Y I Y F Y u u u u u u u Figure 1 Sigle-valued eutrosophic graph Figure 1: Sigle-valued eutrosophic graph The adjacecy matrix of a SVNG give i Figure 1 is The adjacecy matrix of a SVNG give i Fig 1 is AG) =

4 Axioms of 30 The spectrum of a SVNG G give i Figure 1 is as follows: Therefore SpecT Y u j u k )) = { } SpecI Y u j u k )) = { } SpecF Y u j u k )) = { } SpecG) = { } Defiitio 5 The eergy of a SVNG G = X Y) is defied as EG) = ET Y u j u k )) EI Y u j u k )) EF Y u j u k )) = λ j λ j M ζ j N ζ j η j O Defiitio 6 Two SVNGs with the same umber of vertices ad the same eergy are called equieergetic Theorem 1 Let G = X Y) be a SVNG ad AG) be its adjacecy matrix If λ 1 λ 2 λ ζ 1 ζ 2 ζ ad η 1 η 2 η are the eigevalues of AT Y u j u k )) AI Y u j u k )) ad AF Y u j u k )) respectively The η j 1 2 λ j = 0 λ j M ζ j N ζ j = 0 ad η j = 0 η j O λ 2 j = 2 T Y u j u k ) 1 j<k λ j M 2 F Y u j u k ) 1 j<k ζ 2 j = 2 I Y u j u k ) ad 1 j<k ζ j N ηj 2 = η j O Proof 1 Sice AG) is a symmetric matrix whose trace is zero so its eigevalues are real with zero sum 2 By matrix trace properties we have where trat Y u j u k )) ) = λ 2 j λ j M trat Y u j u k )) ) = 0 + T 2 Y u 1u 2 ) + + T 2 Y u 1u )) + T 2 Y u 2u 1 ) T 2 Y u 2u )) + + TY 2u u 1 ) + TY 2u u 2 ) + + 0) = 2 1 j<k T Y u j u k ) Hece λ 2 j = 2 T Y u j u k ) Aalogously we ca show that 1 j<k λ j M ζ 2 j = ζ j N 2 I Y u j u k ) ad ηj 2 = 2 1 j<k η j O 1 j<k F Y u j u k )

5 Axioms of 30 Example 2 Cosider a SVNG G = X Y) o V = {u 1 u 2 u 3 u 4 u 5 u 6 u } as show i Figure 1 The ET Y u j u k )) = 2452 EI Y u j u k )) = ad EF Y u j u k )) = Therefore EG) = Also we have λ j = = 0 λ j M ζ j = = 0 ζ j N η j = = 0 η j O λ j M ζ j N η j O λ 2 j = = 20800) = 2 ζ 2 j = = ) = 2 η 2 j = 8600 = ) = 2 1 j<k 1 j<k 1 j<k T Y u j u k ) I Y u j u k ) F Y u j u k ) We ow give upper ad lower bouds o eergy of a SVNG G i terms of the umber of vertices ad the sum of squares of truth-membership idetermiacy-membership ad falsity-membership values of edges Theorem 2 Let G = X Y) be a SVNG o vertices with adjacecy matrix AG) = AT Y u j u k )) AI Y u j u k )) AF Y u j u k )) The i T Y u j u k ) + 1) T 2 ET Y u j u k )) 2 T Y u j u k ) ii) iii) 1 j<k 2 1 j<k 1 j<k I Y u j u k ) + 1) I 2 EI Y u j u k )) 2 1 j<k I Y u j u k ) 2 F Y u j u k ) + 1) F 2 EF Y u j u k )) 2 F Y u j u k ) 1 j<k 1 j<k where T I ad F are the determiat of AT Y u j u k )) AI Y u j u k )) ad AF Y u j u k )) respectively Proof i) Upper boud: Apply Cauchy-Schwarz iequality to the umbers ad λ 1 λ 2 λ the λ j λ j 2 21) λ j = λ j λ j λ k 22) 1 j<k

6 Axioms of 30 we have By comparig the coefficiets of λ 2 i the characteristic polyomial 1 j<k Substitutig 23) i 22) we obtai Substitutig 24) i 21) we obtai λ λ j ) = AG) λi λ j λ k = 1 j<k T Y u j u k ) 23) λ j 2 = 2 T Y u j u k ) 24) 1 j<k λ j 2 T Y u j u k ) = 1 j<k 2 1 j<k T Y u j u k ) Therefore ET Y u j u k )) 2 1 j<k T Y u j u k ) also sice ii) Lower boud: ET Y u j u k )) = 2 λ j ) = λ j λ j λ k 1 j<k = 2 T Y u j u k ) + 1 j<k Sice AM{ λ j λ k } GM{ λ j λ k } 1 j < k so GM{ λ j λ k } = ET Y u j u k )) λ j λ k 1 j<k 2 1) AM{ λ 2 j λ k } 2 1 j<k T Y u j u k ) + 1)GM{ λ j λ k } 1) = λ j 1 1) = λ j = T 2 so ET Y u j u k )) 2 1 j<k T Y u j u k ) + 1) T 2 Thus 2 T Y u j u k ) + 1) T 2 ET Y u j u k )) 2 T Y u j u k ) 1 j<k 1 j<k

7 Axioms of Aalogously we ca show that 1 j<k 1 j<k I Y u j u k ) ad F Y u j u k ) 2 I Y u j u k ) + 1) I 2 EI Y u j u k )) 1 j<k 2 F Y u j u k ) + 1) F 2 EF Y u j u k )) 1 j<k Example 3 Illustratio to Theorem 2) For the SVNG G give i Figure 1 ET Y u j u k )) = 2452 lower boud = ad upper boud = therefore EI Y u j u k )) = lower boud = ad upper boud = 3041 therefore EF Y u j u k )) = lower boud = ad upper boud = 830 therefore Laplacia Eergy of Sigle-Valued Neutrosophic Graphs I this sectio we defie ad ivestigate the Laplacia eergy of a graph uder sigle-valued eutrosophic eviromet ad ivestigate its properties Defiitio Let G = X Y) be a SVNG o vertices The degree matrix DG) = DT Y u j u k )) DI Y u j u k )) DF Y u j u k )) = [d jk ] of G is a diagoal matrix defied as: d jk = { d G u j ) if j = k 0 otherwise Defiitio 8 The Laplacia matrix of a SVNG G = X Y) is defied as LG) = LT Y u j u k )) LI Y u j u k )) LF Y u j u k )) = DG) AG) where AG) is a adjacecy matrix ad DG) is a degree matrix of a SVNG G Defiitio 9 The spectrum of Laplacia matrix of a SVNG LG) is defied as M L N L O L where M L N L ad O L are the sets of Laplacia eigevalues of LT Y u j u k )) LI Y u j u k )) ad LF Y u j u k )) respectively Example 4 Cosider a SVNG G = X Y) of a graph G = V E) where V = {u 1 u 2 u 3 u 4 u 5 u 6 u } ad E = {u 1 u 4 u 1 u 5 u 2 u 4 u 2 u 5 u 3 u 4 u 3 u 5 u 6 u 4 u 6 u 5 u u 4 u u 5 } as show i Figure 2 u u u u u u Figure 2 Sigle-valued eutrosophic graph Figure 2: Sigle-valued eutrosophic graph u The Laplacia spectrum of a SVNG G give i Fig 2 is Laplacia SpecT Y u j u k )) = { } Laplacia SpecI Y u j u k )) = { }

8 Axioms of 30 LG) = The adjacecy ad the Laplacia matrices of the SVNG show i Figure 2 are as follows: AG) = The Laplacia spectrum of a SVNG G give i Figure 2 is Laplacia SpecT Y u j u k )) = { } Laplacia SpecI Y u j u k )) = { } Laplacia SpecF Y u j u k )) = { } Therefore Laplacia SpecG) = { } Theorem 3 Let G = X Y) be a SVNG ad let LG) = LT Y u j u k )) LI Y u j u k )) LF Y u j u k )) be the Laplacia matrix of G If ϑ 1 ϑ 2 ϑ ϕ 1 ϕ 2 ϕ ad ψ 1 ψ 2 ψ are the eigevalues of LT Y u j u k )) LI Y u j u k )) ad LF Y u j u k )) respectively The 1 2 ϑ j M L ϑ j = 2 1 j<k T Y u j u k ) ϕ j N L ϕ j = 2 1 j<k I Y u j u k ) ad ϑ2 j = 2 T Y u j u k ) + d 2 T 1 j<k Y u j u k ) u j) ϑ j M L d 2 I Y u j u k ) u j) ad F Y u j u k ) + ψ j O L ψ2 j = 2 1 j<k ψ j O L ψ j = 2 1 j<k F Y u j u k ) ϕ2 j = 2 I Y u j u k ) + 1 j<k ϕ j N L d 2 F Y u j u k ) u j) Proof 1 Sice LG) is a symmetric matrix with o-egative Laplacia eigevalues such that ϑ j M L ϑ j = trlg)) = Similarly it is easy to show that ϕ j = 2 ϕ j N L 2 By defiitio of Laplacia matrix we have d TY u j u k )u j ) = 2 1 j<k 1 j<k I Y u j u k ) ad T Y u j u k ) ψ j O L ψ j = 2 1 j<k d TY u j u k )u 1 ) T Y u 1 u 2 ) T Y u 1 u ) T Y u 2 u 1 ) d TY u j u k )u 2 ) T Y u 2 u ) LT Y u j u k )) = T Y u u 1 ) T Y u u 2 ) d TY u j u k )u ) F Y u j u k )

9 Axioms of 30 where By trace properties of a matrix we have trlt Y u j u k )) ) = 2 ϑ j ϑ j M L trlt Y u j u k )) ) = d 2 T Y u j u k ) u 1) + T 2 Y u 1u 2 ) + + T 2 Y u 1u )) +T 2 Y u 2u 1 ) + d 2 T Y u j u k ) u 2) + + T 2 Y u 2u )) + + T 2 Y u u 1 ) + T 2 Y u u 2 ) + + d 2 T Y u j u k ) u )) = 2 1 j<k T Y u j u k ) + d 2 T Y u j u k ) u j) Therefore 2 ϑ j = 2 ϑ j M L 1 j<k T Y u j u k ) + d 2 T Y u j u k ) u j) Aalogously we ca show that ϕ2 j ϕ j N L = 2 I Y u j u k ) + 1 j<k d 2 I Y u j u k ) u j) ad d 2 F Y u j u k ) u j) ψ2 j = 2 F Y u j u k ) + 1 j<k ψ j O L Defiitio 10 The Laplacia eergy of a SVNG G = X Y) is defied as where ϱ j = ϑ j LEG) = LET Y u j u k )) LEI Y u j u k )) LEF Y u j u k )) = ϱ j ξ j 2 T Y u j u k ) 2 I Y u j u k ) 2 F Y u j u k ) 1 j<k 1 j<k 1 j<k ξ j = ϕ j τ j = ψ j Theorem 4 Let G = X Y) be a SVNG ad let LG) be the Laplacia matrix of G If ϑ 1 ϑ 2 ϑ ϕ 1 ϕ 2 ϕ ad ψ 1 ψ 2 ψ are the eigevalues of LT Y u j u k )) LI Y u j u k )) ad LF Y u j u k )) respectively ad ϱ j 2 1 j<k F Y u j u k ) The ϱ j = 0 = ϑ j ϱ 2 j = 2M T 2 T Y u j u k ) 1 j<k ξ j = 0 ξ j = ϕ j τ j = 0 ξ 2 j = 2M I τ 2 j = 2M F τ j 2 I Y u j u k ) 1 j<k τ j = ψ j where M T = 1 j<k T Y u j u k ) d TY u j u k )u j ) 2 1 j<k T Y u j u k ) 2

10 Axioms of 30 M I = M F = 1 j<k I Y u j u k ) j<k F Y u j u k ) d IY u j u k )u j ) d FY u j u k )u j ) 2 1 j<k I Y u j u k ) 2 1 j<k 2 F Y u j u k ) Example 5 Cosider a SVNG G = X Y) o V = {u 1 u 2 u 3 u 4 u 5 u 6 u } as show i Figure 2 The LET Y u j u k )) = 4086 LEI Y u j u k )) = LEF Y u j u k )) = 8229 Therefore LEG) = Also we have ϱ j = 0 ξ j = 0 τ j = 0 ϱ 2 j = = ) = 2M T ξ 2 j = = ) = 2M I τj 2 = = 2661) = 2M F Theorem 5 Let G = X Y) be a SVNG o vertices ad let LG) = LT Y u j u k )) LI Y u j u k )) LF Y u j u k )) be the Laplacia matrix of G The ) i) LET Y u j u k )) 2 T Y u j u k ) + 2 T Y u j u k 1 j<k d TY u j u k )u j ) ; 1 j<k ) ii) LEI Y u j u k )) 2 I Y u j u k ) + 2 I Y u j u k 1 j<k d IY u j u k )u j ) ; 1 j<k ) iii) LEF Y u j u k )) 2 F Y u j u k ) + 2 F Y u j u k 1 j<k d FY u j u k )u j ) 1 j<k Proof Apply Cauchy-Schwarz iequality to the umbers ad ϱ 1 ϱ 2 ϱ we have ϱ j ϱ j 2 2 Sice M T = therefore LET Y u j u k )) 2 LET Y u j u k )) 2M T = 2M T T Y u j u k ) j<k 2 d TY u j u k )u j ) 1 j<k 1 j<k T Y u j u k ) + T Y u j u k ) d TY u j u k )u j ) 2 1 j<k T Y u j u k )

11 Axioms of 30 Aalogously it is easy to show that LEI Y u j u k )) 2 I Y u j u k ) + d IY u j u k )u j ) 1 j<k ad LEF Y u j u k )) 2 F Y u j u k ) + d FY u j u k )u j ) 1 j<k 2 1 j<k ) I Y u j u k 2 1 j<k F Y u j u k ) Theorem 6 Let G = X Y) be a SVNG o vertices ad let LG) = LT Y u j u k )) LI Y u j u k )) LF Y u j u k )) be the Laplacia matrix of G The ) i) LET Y u j u k )) 2 T Y u j u k ) T Y u j u k 1 j<k 2 d TY u j u k )u j ) ; 1 j<k ) ii) LEI Y u j u k )) 2 I Y u j u k ) I Y u j u k 1 j<k 2 d IY u j u k )u j ) ; 1 j<k F Y u j u k iii) LEF Y u j u k )) 2 F Y u j u k ) j<k 2 d FY u j u k )u j ) 1 j<k Proof 2 ϱ j ) = ϱ j ϱ j ϱ k 4M T 1 j<k Sice M T = T Y u j u k ) j<k therefore LET Y u j u k )) 2 1 j<k LET Y u j u k )) 2 M T 2 1 j<k d TY u j u k )u j ) T Y u j u k ) Similarly it is easy to show that LEI Y u j u k )) 2 I Y u j u k ) j<k ad LEF Y u j u k )) 2 F Y u j u k ) j<k T Y u j u k ) 2 1 j<k d TY u j u k )u j ) 2 d IY u j u k )u j ) ) I Y u j u k 1 j<k 2 d FY u j u k )u j ) F Y u j u k ) 1 j<k T Y u j u k ) Theorem Let G = X Y) be a SVNG o vertices ad let LG) = LT Y u j u k )) LI Y u j u k )) LF Y u j u k )) be the Laplacia matrix of G The i) LET Y u j u k )) ϱ 1 + 1) 2 ii) LEI Y u j u k )) ξ 1 + 1) 2 iii) LEF Y u j u k )) τ 1 + 1) 2 1 j<k 1 j<k 1 j<k T Y u j u k ) + I Y u j u k ) + F Y u j u k ) + d TY u j u k )u j ) d IY u j u k )u j ) d FY u j u k )u j ) 2 1 j<k T Y u j u k ) 2 I Y u j u k ) 1 j<k 2 1 j<k F Y u j u k ) ϱ 2 ; 1 ξ 2 ; 1 τ 2 1

12 Axioms of 30 Proof Usig Cauchy-Schwarz iequality we get Sice M T = ϱ j ϱ j j=2 1) ϱ j 2 j=2 ϱ j 2 LET Y u j u k )) ϱ 1 1)2M T ϱ 2 1 ) LET Y u j u k )) ϱ 1 + 1)2M T ϱ 2 1 ) T Y u j u k ) j<k 2 d TY u j u k )u j ) 1 j<k LET Y u j u k )) ϱ 1 + 1) 2 1 j<k T Y u j u k ) + d TYu ju k ) u j) T Y u j u k ) therefore 2 1 j<k T Yu ju k ) ϱ 2 31) 1 Similarly we ca show that LEI Y u j u k )) ξ 1 + 1) 2 1 j<k ad LEF Y u j u k )) τ 1 + 1) 2 1 j<k I Y u j u k ) + F Y u j u k ) + d IY u j u k ) u j) d FY u j u k ) u j) 2 1 j<k I Y u j u k ) 2 1 j<k F Y u j u k ) ξ 2 1 τ 2 1 Theorem 8 If the SVNG G = X Y) is regular the ) i) LET Y u j u k )) ϱ T Y u j u k ) ϱ 2 1 ; 1 j<k ) ii) LEI Y u j u k )) ξ I Y u j u k ) ξ1 2 ; 1 j<k ) iii) LEF Y u j u k )) τ F Y u j u k ) τ1 2 1 j<k Proof Let G be a regular SVNG the 2 T Y u j u k ) 1 j<k d TY u j u k )u j ) = Substitutig 32) i 31) we get LET Y u j u k )) ϱ 1 + 1) 32) 2 T Y u j u k ) ϱ j<k )

13 L + G) = Axioms of 30 ) Similarly it is easy to show that LEI Y u j u k )) ξ I Y u j u k ) ξ1 2 1 j<k ) ad LEF Y u j u k )) τ 1 + 1) 2 F Y u j u k ) τ1 2 1 j<k Theorem 9 Let G = X Y) be a SVNG o vertices with Laplacia matrix LG) = LT Y u j u k )) LI Y u j u k )) LF Y u j u k )) The 4l LET Y u j u k )) = max 2S lt Y u j u k )) 1 l 4l LEI Y u j u k )) = max 2S li Y u j u k )) 1 l 4l LEF Y u j u k )) = max 2S lf Y u j u k )) 1 l 1 j<k 1 j<k 1 j<k T Y u j u k ) I Y u j u k ) F Y u j u k ) where S l T Y u j u k )) = l ϑ j S l I Y u j u k )) = l ϕ j ad S l F Y u j u k )) = 4 Sigless Laplacia Eergy of Sigle-Valued Neutrosophic Graphs l ψ j Defiitio 11 The sigless Laplacia matrix of a SVNG G = X Y) is defied as L + G) = L + T Y u j u k )) L + I Y u j u k )) L + F Y u j u k )) = DG) + AG) where DG) ad AG) are the degree matrix ad the adjacecy matrix respectively of a SVNG G Defiitio 12 The spectrum of sigless Laplacia matrix of a SVNG L + G) is defied as M L + N L + O L + where M L + N L + ad O L + are the sets of sigless Laplacia eigevalues of L + T Y u j u k )) L + I Y u j u k )) ad L + F Y u j u k )) respectively Example 6 Cosider a SVNG G = X Y) of a graph G = V E) where V = {u 1 u 2 u 3 u 4 u 5 u 6 u } ad E = {u 1 u 2 u 2 u 3 u 3 u 4 u 4 u 5 u 5 u u 6 u u 6 u 1 u 3 u 5 u 1 u u 2 u u 3 u } as show i Figure 3 u u u u u u u Figure 3 3: Sigle-valued eutrosophic graph The adjacecy ad the sigless Laplacia matrices AG) of = the SVNG show i Figure 3 are as follows:

14 Axioms of 30 AG) = L + G) = The sigless Laplacia spectrum of a SVNG G give i Figure 3 is Sigless Laplacia SpecT Y u j u k )) = { } Sigless Laplacia SpecI Y u j u k )) = { } Sigless Laplacia SpecF Y u j u k )) = { } Therefore Sigless Laplacia SpecG) = { } Theorem 10 Let G = X Y) be a SVNG ad let L + G) be the sigless Laplacia matrix of G If ϑ + 1 ϑ+ 2 ϑ+ ϕ + 1 ϕ + 2 ϕ + ad ψ + 1 ψ + 2 ψ + are the eigevalues of L + T Y u j u k )) L + I Y u j u k )) ad L + F Y u j u k )) respectively The 1 2 ϑ + j = 2 T Y u j u k ) 1 j<k ϑ + j M L + 2 F Y u j u k ) 1 j<k ϕ + j N L + ϕ + j = 2 I Y u j u k ) ad 1 j<k ψ + j = ψ + j O L + ϑ + j = 2 T Y u j u k ) + d 2 T 1 j<k Y u j u k ) u j) ϕ + j = 2 I Y u j u k ) + 1 j<k ϑ + j M L + ϕ + j N L + d 2 I Y u j u k ) u j) ad ψ + j = 2 F Y u j u k ) + d 2 F 1 j<k Y u j u k ) u j) ψ + j O L + Proof Proof follows at oce from proof of Theorem 3 Defiitio 13 The sigless Laplacia eergy of a SVNG G = X Y) is defied as where LE + G) = LE + T Y u j u k )) LE + I Y u j u k )) LE + F Y u j u k )) = ϱ + j = ϑ + j ϱ + j ξ + j τ + j 2 T Y u j u k ) 2 I Y u j u k ) 2 F Y u j u k ) 1 j<k ξ + j = ϕ + 1 j<k j τ + j = ψ + 1 j<k j

15 Axioms of 30 Theorem 11 Let G = X Y) be a SVNG ad let L + G) be the sigless Laplacia matrix of G If ϑ + 1 ϑ + 2 ϑ + ϕ + 1 ϕ + 2 ϕ + ad ψ + 1 ψ + 2 ψ + are the eigevalues of L + T Y u j u k )) L + I Y u j u k )) ad L + F Y u j u k )) respectively ad ϱ + j = ϑ + j ϕ + j 2 I Y u j u k ) 2 1 j<k τ + j = ψ + j ϱ + j = 0 F Y u j u k ) 1 j<k ξ + j = 0 The τ + j = 0 ϱ + j = 2M + T ξ + j = 2M + I τ + j = 2M + F 2 T Y u j u k ) 1 j<k ξ + j = where M + T = 1 j<k T Y u j u k ) M + I = 1 j<k I Y u j u k ) M + F = 1 j<k F Y u j u k ) d TY u j u k )u j ) d IY u j u k )u j ) d FY u j u k )u j ) 2 1 j<k 2 1 j<k T Y u j u k ) 2 1 j<k I Y u j u k ) 2 F Y u j u k ) 2 2 Example Cosider a SVNG G = X Y) o V = {u 1 u 2 u 3 u 4 u 5 u 6 u } as show i Figure 3 The LE + T Y u j u k )) = LE + I Y u j u k )) = LE + F Y u j u k )) = Therefore LE + G) = Also we have ϱ + j = 0 ξ + j = 0 τ + j = 0 ϱ + j = = 21311) = 2M + T ξ + j = = 20143) = 2M + I τ + j = = ) = 2M + F Theorem 12 Let G = X Y) be a SVNG o vertices with sigless Laplacia matrix L + G) = L + T Y u j u k )) L + I Y u j u k )) L + F Y u j u k )) The 4l LE + T Y u j u k )) = max 2S+ l T Y u j u k )) 1 l 4l LE + I Y u j u k )) = max 2S+ l I Y u j u k )) 1 l 1 j<k 1 j<k T Y u j u k ) I Y u j u k )

16 Axioms of 30 4l LE + F Y u j u k )) = max 2S+ l F Y u j u k )) 1 l 1 j<k F Y u j u k ) where S + l T Y u j u k )) = l ϑ + j S + l I Y u j u k )) = l ϕ + j ad S + l F Y u j u k )) = l ψ + j 5 Relatio amog Eergy Laplacia Eergy ad Sigless Laplacia Eergy of SVNGs This sectio discusses the relatioship amog eergy Laplacia eergy ad sigless Laplacia eergy of SVNGs Theorem 13 Let G be a SVNG o vertices ad let AG) LG) ad L + G) be the adjacecy the Laplacia ad the sigless Laplacia matrices of G respectively The LE + G) LEG) 2EG) Proof Clearly L + T Y u j u k )) LT Y u j u k )) 2 T Y u j u k ) 2 T Y u j u k ) 1 j<k 1 j<k = DT Y u j u k )) + AT Y u j u k )) 2 T Y u j u k ) 2 T Y u j u k ) 1 j<k 1 j<k = DT Y u j u k )) AT Y u j u k )) 51) 52) From Equatios 51) ad 52) we get L + T Y u j u k )) 2 1 j<k T Y u j u k ) LT Y u j u k )) 2 1 j<k T Y u j u k ) = 2AT Y u j u k )) The LT Y u j u k )) 2 1 j<k T Y u j u k ) = L + T Y u j u k )) 2 1 j<k T Y u j u k ) 2AT Y u j u k )) Also L + T Y u j u k )) 2 1 j<k T Y u j u k ) = 2AT Y u j u k )) + LT Y u j u k )) 2 1 j<k T Y u j u k ) By well kow property of eergy of a graph ) 2 T Y u j u k ) 1 j<k LET Y u j u k )) = E LT Y u j u k )) E L + T Y u j u k )) +E 2AT Y u j u k ))) = LE + T Y u j u k )) + 2ET Y u j u k )) ) 2 LE + T Y u j u k )) = E L + T Y u j u k ) 1 j<k T Y u j u k )) E LT Y u j u k )) +E2AT Y u j u k ))) = LET Y u j u k )) + 2ET Y u j u k )) 2 1 j<k 2 1 j<k ) T Y u j u k ) ) T Y u j u k ) 53) 54)

17 Axioms of 30 Combiig 53) ad 54) we get LE + T Y u j u k )) LET Y u j u k )) 2ET Y u j u k )) Aalogously we ca show that LE + I Y u j u k )) LEI Y u j u k )) 2EI Y u j u k )) ad LE + F Y u j u k )) LEF Y u j u k )) 2EF Y u j u k )) Hece LE + G) LEG) 2EG) Theorem 14 If the SVNG G is regular The EG) = LEG) = LE + G) Theorem 15 Let G = X Y) be a SVNG o vertices ad let LG) ad L + G) be the Laplacia ad the sigless Laplacia matrices of G respectively The { LE + T Y u j u k )) + LET Y u j u k )) max 2ET Y u j u k )) 2 d T Y u j u k )u j ) { LE + I Y u j u k )) + LEI Y u j u k )) max 2EI Y u j u k )) 2 d I Y u j u k )u j ) { LE + F Y u j u k )) + LEF Y u j u k )) max 2EF Y u j u k )) 2 d F Y u j u k )u j ) 2 1 j<k 2 1 j<k 2 1 j<k T Y u j u k ) I Y u j u k ) F Y u j u k ) } } } Theorem 16 Let G = X Y) be a SVNG o vertices ad let LG) ad L + G) be the Laplacia ad the sigless Laplacia matrices of G respectively The 4r T Y u j u k ) LE + 1 j<k T Y u j u k )) + LET Y u j u k )) 4ET Y u j u k )) 4r I Y u j u k ) LE + 1 j<k I Y u j u k )) + LEI Y u j u k )) 4EI Y u j u k )) 4r F Y u j u k ) LE + 1 j<k F Y u j u k )) + LEF Y u j u k )) 4EF Y u j u k )) where r is the umber of o-zero eigevalues of SVNG G Theorem 1 Let G = X Y) be a SVNG o vertices ad let LG) = LT Y u j u k )) LI Y u j u k )) LF Y u j u k )) be the Laplacia matrix of G The LET Y u j u k )) ET Y u j u k )) + d TY u j u k ) u j) LEI Y u j u k )) EI Y u j u k )) + d IY u j u k ) u j) LEF Y u j u k )) EF Y u j u k )) + d FY u j u k ) u j) 2 1 j<k 2 1 j<k 2 1 j<k T Y u j u k ) I Y u j u k ) F Y u j u k ) Theorem 18 Let G = X Y) be a SVNG o vertices ad let L + G) = L + T Y u j u k )) L + I Y u j u k )) L + F Y u j u k )) be the sigless Laplacia matrix of G The LE + T Y u j u k )) ET Y u j u k )) + d TY u j u k ) u j) LE + I Y u j u k )) EI Y u j u k )) + d IY u j u k ) u j) 2 1 j<k 2 1 j<k T Y u j u k ) I Y u j u k )

18 Axioms of 30 LE + F Y u j u k )) EF Y u j u k )) + d FY u j u k ) u j) 2 1 j<k F Y u j u k ) Theorem 19 Let G be a coected SVNG o vertices ad let L + G) = L + T Y u j u k )) L + I Y u j u k )) L + F Y u j u k )) be the sigless Laplacia matrix of G The LE + T Y u j u k )) ET Y u j u k )) + LE + I Y u j u k )) EI Y u j u k )) + LE + F Y u j u k )) EF Y u j u k )) + d 2 T Y u j u k ) u j) 4 d 2 I Y u j u k ) u j) 4 d 2 F Y u j u k ) u j) 4 6 Applicatio of Eergy of SVNGs i Group Decisio-Makig 1 j<k 1 j<k 1 j<k 2 T Y u j u k ) 2 I Y u j u k ) 2 F Y u j u k ) Group decisio-makig is a commoly used tool i huma activities which determies the optimal alterative from a give fiite set of alteratives usig the evaluatio iformatio give by a group of decisio makers or experts With the rapid developmet of society group decisio-makig plays a icreasigly importat role whe dealig with the decisio-makig problems Recetly may scholars have ivestigated the approaches for group decisio-makig based o differet kids of decisio iformatio However i order to reflect the relatioships amog the alteratives we eed to make pairwise comparisos for all the alteratives i the process of decisio- makig Preferece relatio is a powerful quatitative decisio techique that support experts i expressig their prefereces over the give alteratives For a set of alteratives Z = {z 1 z 2 z } the experts compare each pair of alteratives ad costruct preferece relatios respectively If every elemet i the preferece relatios is a sigle-valued eutrosophic umber the the cocept of the sigle-valued eutrosophic preferece relatio SVNPR) ca be put forth as follows: Defiitio 14 A SVNPR o the set Z = {z 1 z 2 z } is represeted by a matrix R = r jk ) where r jk = z j z k Tz j z k ) Iz j z k ) Fz j z k ) for all j k = 1 2 For coveiece let r jk = T jk I jk F jk where T jk idicates the degree to which the object z j is preferred to the object z k F jk deotes the degree to which the object z j is ot preferred to the object z k ad I jk is iterpreted as a idetermiacy-membership degree with the coditios: T jk I jk F jk [0 1] T jk = F kj F jk = T kj I jk + I kj = 1 T jj = I jj = F jj = 05 for all j k = 1 2 A group decisio-makig problem cocerig the Alliace parter selectio of a software compay is solved to illustrate the applicability of the proposed cocepts of eergy of SVNGs i realistic sceario Alliace Parter Selectio of a Software Compay Eastsoft is oe of the top five software compaies i Chia [41] It offers a rich portfolio of busiesses icludig product egieerig solutios idustry solutios ad related software products ad platform ad services It is dedicated to becomig a globally leadig IT solutios ad services provider through cotiuous improvemet of orgaizatio ad process competece developmet of leadership ad employees ad alliace ad ope iovatio To improve the operatio ad competitiveess capability i the global market Eastsoft plas to establish a strategic alliace with

19 Axioms of 30 a trasatioal corporatio After umerous cosultatios five trasatioal corporatios would like to establish a strategic alliace with Eastsoft; they are HP a 1 PHILIPS a 2 EMC a 3 SAP a 4 ad LK a 5 To select the desirable strategic alliace parter three experts e i i = 1 2 3) are ivited to participate i the decisio aalysis who come from the egieerig maagemet departmet the huma resources departmet ad the fiace departmet of Eastsoft respectively Based o their experieces the experts compare each pair of alteratives ad give idividual judgmets usig the followig SVNPRs R i = r i) jk ) 5 5 i = 1 2 3): The SVNDGs D i correspodig to SVNPRs R i i = 1 2 3) give i Tables 1 3 are show i Figure 4 Table 1 SVNPR of the expert from the egieerig maagemet departmet R 1 a 1 a 2 a 3 a 4 a 5 a a a a a Table 2 SVNPR of the expert from the huma resources departmet R 2 a 1 a 2 a 3 a 4 a 5 a a a a a Table 3 SVNPR of the expert from the fiace departmet R 3 a 1 a 2 a 3 a 4 a 5 a a a a a The eergy of a SVNDG is the sum of absolute values of the real part of eigevalues of D The eergy of each SVNDG D i i = 1 2 3) is calculated as: ED 1 ) = ED 2 ) = ED 3 ) = The the weight of each expert ca be determied as: w i = w T ) i w I ) i w F ) i ) = ED T ) i ) m ED T ) l ) l=1 ED I ) i ) m ED I ) l ) l=1 ED F ) i ) i = 1 2 m ED F ) l ) w 1 = w 2 = w 3 = Utilize the aggregatio operator to fuse all the idividual SVNPRs R i = r i) jk ) 5 5 i = 1 2 3) ito the collective SVNPR R = r jk ) 5 5 as show i Table 4 Here we apply the sigle-valued eutrosophic weighted averagig SVNWA) operator [42] to fuse the idividual SVNPR SVNWAr 1) jk r2) jk rs) jk ) = s 1 1 T i) jk i=1 m l=1 ) wi s i=1 I i) ) wi s jk i=1 F i) ) wi jk

20 Axioms of 30 Table 4 The collective SVNPR of all the above idividual SVNPRs R a 1 a 2 a 3 a 4 a 5 a a a a a a5 a a a a5 a a a a a D 1 D a5 a a a a D 3 Figure 4 Sigle-valued eutrosophic digraphs Draw a directed etwork correspodig to a collective SVNPR above as show i Figure 5 The uder the coditio T jk 05 j k = ) a partial diagram is draw as show i Figure a 5 a 2 a a a Figure 5 Directed etwork of the fused SVNPR

21 Axioms of a a a 3 a a Figure 6 Partial directed etwork of the fused SVNPR Calculate the out-degrees out-da j ) j = ) of all criteria i a partial directed etwork as follows: out-da 1 ) = out-da 2 ) = out-da 3 ) = out-da 4 ) = out-da 5 ) = Accordig to membership degrees of out-da j ) j = ) we get the rakig of the factors a j j = ) as: a 5 a 3 a 2 a 1 a 4 Thus the best choice is LK a 5 Now elemets of the Laplacia matrices of the SVNDGs LD i ) = Ri L Figure 4 are provided i Tables 5 i = 1 2 3) show i Table 5 Elemets of the Laplacia matrix of the SVNDG D 1 R L 1 a 1 a 2 a 3 a 4 a 5 a a a a a Table 6 Elemets of the Laplacia matrix of the SVNDG D 2 R L 2 a 1 a 2 a 3 a 4 a 5 a a a a a

22 Axioms of 30 Table Elemets of the Laplacia matrix of the SVNDG D 3 R L 3 a 1 a 2 a 3 a 4 a 5 a a a a a The Laplacia eergy of each SVNDG is calculated as: LED 1 ) = LED 2 ) = LED 3 ) = The the weight of each expert ca be determied as: w i = w T ) i w I ) i w F ) i ) = LED T ) i ) LED I ) i ) m LED T ) l ) LED I ) l ) l=1 m l=1 LED F ) i ) i = 1 2 m LED F ) l ) w 1 = w 2 = w 3 = based o which usig the SVNWA operator the fused SVNPR is determied as show i Table 8 m l=1 Table 8 The collective SVNPR of all the above idividual SVNPRs R a 1 a 2 a 3 a 4 a 5 a a a a a I the directed etwork correspodig to a collective SVNPR above we select those sigle-valued eutrosophic umbers whose membership degrees T jk 05 j k = ) ad resultig partial diagram is show i Figure a 5 a a a a Figure Partial directed etwork of the fused SVNPR

23 Axioms of 30 Calculate the out-degrees out-da j ) j = ) of all criteria i a partial directed etwork as follows: out-da 1 ) = out-da 2 ) = out-da 3 ) = out-da 4 ) = out-da 5 ) = Accordig to membership degrees of out-da j ) j = ) we get the rakig of the factors a j j = ) as: a 5 a 3 a 2 a 1 a 4 Thus the best choice is LK a 5 Now elemets of the sigless Laplacia matrices of the SVNDGs L + D i ) = Ri L+ i = 1 2 3) show i Figure 4 are give i Tables 9 11 Table 9 Elemets of the sigless Laplacia matrix of the SVNDG D 1 R L+ 1 a 1 a 2 a 3 a 4 a 5 a a a a a Table 10 Elemets of the sigless Laplacia matrix of the SVNDG D 2 R L+ 2 a 1 a 2 a 3 a 4 a 5 a a a a a Table 11 Elemets of the sigless Laplacia matrix of the SVNDG D 3 R L+ 3 a 1 a 2 a 3 a 4 a 5 a a a a a The sigless Laplacia eergy of each SVNDG is calculated as: LE + D 1 ) = LE + D 2 ) = LE + D 3 ) = The the weight of each expert is w i = w T ) i w I ) i w F ) i ) = LE + D T ) i ) LE + D I ) i ) LE + m D T ) l ) LE + D I ) l ) l=1 m l=1 LE + D F ) i ) LE + i = 1 2 m D F ) l ) w 1 = w 2 = w 3 = based o which fuse all the idividual SVNPRs R i = r i) jk ) 5 5 i = 1 2 3) ito the collective SVNPR R = r jk ) 5 5 by usig the SVNWA operator as show i Table 12 m l=1

24 Axioms of 30 Table 12 The collective SVNPR of all the above idividual SVNPRs R a 1 a 2 a 3 a 4 a 5 a a a a a I the directed etwork correspodig to a collective SVNPR above we select those sigle-valued eutrosophic umbers whose membership degrees T jk 05 j k = ) ad resultig partial diagram is show i Figure a 5 a a a a Figure 8 Partial directed etwork of the fused SVNPR Calculate the out-degrees out-da j ) j = ) of all criteria i a partial directed etwork as follows: out-da 1 ) = out-da 2 ) = out-da 3 ) = out-da 4 ) = out-da 5 ) = Accordig to membership degrees of out-da j ) j = ) we get the rakig of the factors a j j = ) as: a 5 a 3 a 2 a 1 a 4 Thus the best choice is LK a 5 Real Time Example I this sectio the proposed cocepts of eergy Laplacia eergy ad sigless Laplacia eergy of a SVNG are explaied through a real time example We have take the Website modeled as a SVNG by cosiderig the avigatio of the customer We have take the four liks: 1 microcotroller-boards 2 log-i html 3 ad 4 project kits for our calculatio A SVNG of this site for four differet time periods is cosidered The eergy Laplacia eergy ad sigless Laplacia eergy of a SVNG is calculated for each of these periods The eergy Laplacia eergy ad sigless Laplacia eergy are represeted i terms of bar graphs I the website accessed o 8 May 2012) the above 4 liks are cosidered for the period 16 Jauary 2018 to 15 February 2018 ad for this graph as show i Figure 9 we have

25 x Axioms of 30 SpecT Y u j u k )) = { } SpecI Y u j u k )) = { } x 4 SpecF Y u j u k )) = { } ET Y u j u k )) = EI Y u j u k )) = 1844 EF Y u j u k )) = Therefore EG 1 ) = Laplasia SpecT Y u j u k )) = { } Laplacia SpecI Y u j u k )) = { } Laplacia SpecF Y u j u k )) = { } LET Y u j u k )) = LEI Y u j u k )) = LEF Y u j u k )) = 2290 Therefore LEG 1 ) = LE + T Y u j u k )) = LE + I Y u j u k )) = LE + F Y u j u k )) = Therefore LE + G 1 ) = x Figure 8: Partial directed etwork of the fused SVNPR modeled as a SVNG by cosiderig the avigatio of the customer We have take the four liks: 1 microcotroller-boards 2/log-i html 3/ ad 4/ project kits for our calculatio A SVNG of this site for four differet time periods is cosidered The eergy Laplacia eergy ad sigless Laplacia eergy Sigless Laplacia SpecT Y u j u k )) = { } of a SVNG is calculated for each of these periods The eergy Laplacia eergy ad sigless Laplacia Sigless Laplacia SpecI Y u j u k )) = { } eergy are represeted Sigless Laplacia i SpecF terms Y u of j ubar k )) = graphs { I the website10211} the above 4 liks are cosidered for the period Jauary to February ad for this graph we have u u u u Figure 9 Sigle-valued eutrosophic graph G 1 Figure 9: SVNG G 1 For the period 16 February 2018 to 15 March 2018 see Figure 10) we have SpecT Y u j u k )) = { } SpecI Y u j uspect k )) = Y { u j u k )) = { } 0544} SpecF Y u j uspeci k )) = Y u { 0603 j u k )) = { } 1313} ET Y u j u k )) SpecF = Y u j u k )) EI = { 0503 Y u j u k )) = EF 08430} Y u j u k )) = Therefore EG ET 1 Y )u = j u k )) = EI Y u19999 j u k )) = 2418 EF Y u j u k )) = 1684 Therefore EG 2 ) = Laplasia SpecT Y u j u k )) = { } Laplacia SpecT Laplacia SpecI Y u j u k )) Y u = j u {0 k )) = { } } Laplacia SpecI Laplacia SpecF Y u j u k )) Y u = j u {0 k )) = { } } Laplacia SpecF Y u j u k )) = { } LET Y u j u k LET )) = LEI Y u j u k )0051 LEF Y u j u k )) = 2290 Y u j u k )) = 136 LEI Y u j u k )) = LEF Y u j u k )) = 2065 Therefore LEG 1 ) = Therefore LEG 2 ) = Sigless Laplacia SpecT Y u j u k )) = { } Sigless Laplacia SpecI Y u j u k )) = { } Sigless Laplacia SpecF Y u j u k )) = { } LE + T Y u j u k )) = 1306 LE + I Y u j u k )) = LE + F Y u j u k )) = Therefore LE + G 2 ) =

26 LE + T Y u j u k )) = LE + I Y u j u k )) = LE + F Y u j u k )) = Therefore LE + G 1 ) = For the period February to March we have Axioms of 30 u u u u Figure 10 Sigle-valued eutrosophic graph G 2 Figure 10: SVNG G 2 For the period 16 March 2018 to 15 April 2018 see Figure 11) we have SpecT Y u j u k )) = { } SpecI Y u j u k SpecT )) = Y { 0909 u j u k )) = { } 10168} SpecF Y u j uspeci k )) = Y { 0503 u j u k )) = { } 156} ET Y u j u k )) SpecF = Y u j u k )) EI = { Y u j u k )) = EF 1283} Y u j u k )) = 1684 Therefore EG ET 2 Y )u = j u k )) = EI Y u1684 j u k )) = EF Y u j u k )) = Therefore EG 3 ) = Laplacia SpecT Y u j u k )) = { } Laplacia SpecT Y u j u k )) = { } Laplacia SpecI Laplacia Y u j SpecI u k )) Y = u j {0 u } k )) = { } Laplacia SpecF Laplacia Y uspecf j u k )) Y = u j {0 u k )) 0562 = { } 2309} LET Y u j u k )) LET = Y 136 u j u k )) LEI = 2892 Y ulei j u k Y )) u= j u k )) = 3056LEF Y u u j u jk u)) k = )) = 2065 Therefore LEG Therefore 2 ) = LEG ) = Sigless Laplacia SpecT Y u j u k )) = { } Sigless Laplacia SpecT Y u j u k )) = { } Sigless Laplacia SpecI Sigless Laplacia SpecI Y u j u Y u k )) = j u k )) = { } { } Sigless Laplacia SpecF Sigless Laplacia SpecF Y u j u Y u k )) = j u k )) = { } { } LE + LE + T Y u j u k )) = 2461 T Y u j u k )) = 1306 LE + LE + I Y u j u k )) = LE + F I Y u j u k )) = LE + Y u j u k )) = F Y u j u k )) = Therefore Therefore LE + LE + G 3 ) = G 2 ) = u For the period March to April we have u u u Figure 11 Sigle-valued eutrosophic graph G 3 Figure 11: SVNG G 3 Fially for the period 16 April 2018 to 15 May 2018 see Figure 12) we have SpecT Y u j u k )) = { } SpecI Y u j u k )) = { } 2 SpecF Y u j u k )) = { } ET Y u j u k )) = EI Y u j u k )) = EF Y u j u k )) = Therefore EG 3 ) = Laplacia SpecT Y u j u k )) = { }

27 Figure 11: SVNG G 3 SpecT Y u j u k )) = { } SpecI Y u j u k )) = { } SpecF Y u j u k )) = { } ET Y u j u k )) = EI Y u j u k )) = EF Y u j u k )) = Therefore EG SpecT 3 ) = Y u j u k )) = { } Laplacia SpecT SpecI Y u j u jk u)) k )) = { 1088 = { } 115} Laplacia SpecI SpecF Y u j u k )) k )) = = { 0686 { } 10680} Laplacia SpecF ET Y u j u k )) = 1338EI Y u j u k )) = 33265EF Y u j u k )) = Y u j u k )) = { } Therefore EG 4 ) = LET Y u j u k )) = 2892 LEI Y u j u k )) = 3056LEF Y u j u k )) = Therefore LEG Laplacia 3 ) = SpecT 2892 Y u j u3056 k )) = { } Sigless Laplacia SpecT SpecI Y u u j u jk u)) k = )){0 = { } } Sigless Laplacia SpecI SpecF Y u j u j k u)) k )) = {0 = { } } LET Y u j u k )) = 1425 LEI Y u j u k )) = 3868 LEF Y u j u k )) = Sigless Laplacia SpecF Y u j u k )) = { } Therefore LEG 4 ) = LE + T Y u j u k )) = 2461 LE + I Y u j u k )) = LE + F Y u j u k )) = Therefore LE + G 3 ) = Axioms of 30 Sigless Laplacia SpecT Y u j u k )) = { } Sigless Laplacia SpecI Y u j u k )) = { } Fially Sigless for thelaplacia periodspecf April Y u16 j u k )018 = { 0996 to May } we have LE + T Y u j u k )) = LE + I Y u j u k )) = LE + F Y u j u k )) = Therefore LE + G 4 ) = u u u u Figure 12 Sigle-valued eutrosophic graph G 4 Figure 12: SVNG G 4 SpecT Y u j u k )) = { Idetermiacy membership 06662} 3 Falsity membership SpecI Y u j u k )) = { } SpecF Y u j u k )) = { } ET Y u j u k )) = 1338EI Y u j u k )) = 33265EF Y u j u k )) = Therefore EG 4 ) = Truth membership Ja Feb Feb Mar Mar April April May Figure 13 Eergy of sigle-valued eutrosophic graphs

28 Axioms of Truth membership Idetermiacy membership Falsity membership Ja Feb Feb Mar Mar April April May Figure 14 Laplacia eergy of sigle-valued eutrosophic graphs 4 35 Truth membership Idetermiacy membership Falsity membership Ja Feb Feb Mar Mar April April May Figure 15 Sigless Laplacia eergy of sigle-valued eutrosophic graphs The bar graphs show i Figures represet the eergy Laplacia eergy ad sigless Laplacia eergy of four liks for the above four periods correspodig to the truth-membership idetermiacy-membership ad falsity-membership values From the above bar graphs the eergy Laplacia eergy ad sigless Laplacia eergy of truth-membership for the period March to April is high as compared to other periods the eergy Laplacia eergy ad sigless Laplacia eergy of idetermiacy-membership for the period April to May is high ad the eergy Laplacia eergy ad sigless Laplacia eergy of falsity-membership for the period March to April is high 8 Coclusios A sigle-valued eutrosophic model is used i computer techology etworkig commuicatio whe the cocept of idetermiacy is preset I this paper we have itroduced certai ovel cocepts icludig eergy Laplacia eergy ad sigless Laplacia eergy of SVNGs We have derived the lower ad upper bouds for the eergy ad Laplacia eergy of a SVNG We have obtaied the relatios amog eergy Laplacia eergy ad sigless Laplacia eergy of a SVNG Amog the properties of eergy Laplacia eergy ad sigless Laplacia eergy of a SVNG there is a great deal of aalogy but also some sigificat differeces Fially applicatio i group decisio-makig based o SVNPRs is preseted to illustrate the applicability of the proposed cocepts of SVNGs These cocepts

29 Axioms of 30 are also illustrated with real time example We are plaig to exted our research work to 1) Eergy of bipolar eutrosophic graphs; 2) Simplified iterval-valued Pythagorea fuzzy graphs; 3) Hesitat Pythagorea fuzzy graphs; Eergy of eutrosophic hypergraphs Author Cotributios: SN MA ad FS coceived ad desiged the experimets; MA ad FS aalyzed the data; SN wrote the paper Coflicts of Iterest: The authors declare o coflict of iterest Refereces 1 Smaradache F A uifyig field i logics I Neutrosophy: Neutrosophic Probability Set ad Logic; America Research Press: Rehoboth DE USA Wag H; Smaradache F; Zhag YQ; Suderrama R Sigle-valued eutrosophic sets Multisp Multistruct Ataassov KT Ituitioistic fuzzy sets Fuzzy Sets Syst Bhattacharya S Neutrosophic iformatio fusio applied to the optios market Ivest Maag Fiac Iov Aggarwal S; Biswas R; Asari AQ Neutrosophic modelig ad cotrol I Prroceedigs of the 2010 Iteratioal Coferece o Computer ad Commuicatio Techology Allahabad Idia 1 19 September 2010; pp Guo Y; Cheg HD New eutrosophic approach to image segmetatio Patter Recogit Ye J; Fu J Multi-period medical diagosis method usig a sigle-valued eutrosophic similarity measure based o taget fuctio Comput Methods Programs Biomed Ye J Multicriteria decisio-makig method usig the correlatio coefficiet uder sigle-valued eutrosophic eviromet It J Ge Syst Naz S; Rashmalou H; Malik MA Operatios o sigle-valued eutrosophic graphs with applicatio J Itell Fuzzy Syst Ashraf S; Naz S; Rashmalou H; Malik MA Regularity of graphs i sigle-valued eutrosophic eviromet J Itell Fuzzy Syst Gutma I The eergy of a graph Ber Math Stat Sekt Forsch Graz Gutma I The Eergy of a Graph: Old ad New Results Algebraic Combiatorics ad Applicatios; Spriger: Berli/Heidelberg Germay 2001; pp Gutma I; Zhou B Laplacia eergy of a graph I Liear Algebra ad Its Applicatios; Elsevier: New York NY USA 2006; Volume 414 pp Gutma I; Robbiao M; Martis EA; Cardoso DM; Media L; Rojo O Eergy of lie graphs I Liear Algebra ad Its Applicatios; Elsevier: New York NY USA 2010; Volume 433 pp Erdos P Graph theory ad probability caad J Math Zadeh LA Fuzzy sets If Cotrol Kaufma A Itroductio a la Theorie des Sour-Esembles Flous; Masso et Cie: Paris Frace 193; p 1 18 Rosefeld A Fuzzy Graphs Fuzzy Sets ad Their Applicatios; Zadeh LA Fu KS Shimura M Eds; Academic Press: New York NY USA 195; pp Ajali N; Mathew S Eergy of a fuzzy graph A Fuzzy Maths Iform Sharbaf SR; Fayazi F Laplacia eergy of a fuzzy graph Ira J Math Chem Parvathi R; Karuambigai MG Ituitioistic fuzzy graphs I Computatioal Itelligece Theory ad Applicatios; Spriger: Berli/Heidelberg Germay 2006; pp Akram M Bipolar fuzzy graphs If Sci Akram M; Davvaz B Strog ituitioistic fuzzy graphs Filomat Akram M; Malik HM; Shahzadi S; Smaradache F Neutrosophic soft rough graphs with applicatio Axioms Akram M; Siddique S Neutrosophic competitio graphs with applicatios J Itell Fuzzy Syst Akram M; Sitara M; Smaradache F Graph structures i bipolar eutrosophic eviromet Mathematics

30 Axioms of 30 2 Naz S; Ashraf S; Akram M A ovel approach to decisio-makig with Pythagorea fuzzy iformatio Mathematics Ishfaq N; Sayed S; Akram M; Smaradache F Notios of rough eutrosophic digraphs Mathematics Sayed S; Ishfaq N; Akram M; Smaradache F Rough eutrosophic digraphs with applicatio Axioms Akram M; Ishfaq N; Sayed S; Smaradache F Decisio-makig approach based o eutrosophic rough iformatio Algorithms Naz S; Malik MA; Rashmalou H Hypergraphs ad trasversals of hypergraphs i iterval-valued ituitioistic fuzzy settig J Mult-Valued Logic Soft Comput Praba B; Chadrasekara VM; Deepa G Eergy of a itutioistic fuzzy graph Ital J Pure Appl Math Basha SS; Kartheek E Laplacia eergy of a ituitioistic fuzzy graph Id J Sci Techol Broumi S; Smaradache F; Talea M; Bakali A Sigle-valued eutrosophic graphs: degree order ad size I Proceedigs of the 2016 IEEE Iteratioal Coferece o Fuzzy Systems Vacouver BC Caada July 2016; pp Broumi S; Bakali A; Talea M; Smaradache F; Kishore KK; Sahi R Shortest path problem uder iterval valued eutrosophic settig J Fudam Appl Sci Hamidi M; Saeid AB Accessible sigle-valued eutrosophic graphs J Appl Math Comput [CrossRef] 3 Staujkic D; Zavadskas EK; Smaradache F; Brauers WK; Karabasevic D A eutrosophic extesio of the MULTIMOORA method Iformatica Smaradache F; Ye J Summary of the Special Issue Neutrosophic Iformatio Theory ad Applicatios at Iformatio Joural Iformatio Tripathi G A matrix extesio of the Cauchy-Schwarz iequality Eco Lett Akram M; Shahzadi S Neutrosophic soft graphs with applicatio J Itell Fuzzy Syst Fa ZP; Liu Y A approach to solve group decisio-makig problems with ordial iterval umbers IEEE Tras Syst Ma Cyber B Cyber Biswas P; Pramaik S; Giri BC TOPSIS method for multi-attribute group decisio-makig uder sigle-valued eutrosophic eviromet Neural Comput Appl c 2018 by the authors Licesee MDPI Basel Switzerlad This article is a ope access article distributed uder the terms ad coditios of the Creative Commos Attributio CC BY) licese