Properties of SuperHyperGraph and Neutrosophic SuperHyperGraph

Neutrosophic Sets and Systems, Vol. 49, 2022

PropertiesofSuperHyperGraphandNeutrosophic SuperHyperGraph

HenryGarrett1,∗

1DepartmentofMathematics,PayameNoorUniversity,P.O.Box:19395-3697,Tehran,Iran; DrHenryGarrett@gmail.com

∗Correspondence:DrHenryGarrett@gmail.com

Abstract Newsettingisintroducedtostudydominating,resolving,coloring,Eulerian(Hamiltonian)neutrosophicpath,n-Eulerian(Hamiltonian)neutrosophicpath,zeroforcingnumber,zeroforcingneutrosophicnumber,independentnumber,independentneutrosophic-number,cliquenumber,cliqueneutrosophic-number, matchingnumber,matchingneutrosophic-number,girth,neutrosophicgirth,1-zero-forcingnumber,1-zeroforcingneutrosophic-number,failed1-zero-forcingnumber,failed1-zero-forcingneutrosophic-number,globaloffensivealliance,t-offensivealliance,t-defensivealliance,t-powerfulalliance,andglobal-powerfulalliancein SuperHyperGraphandNeutrosophicSuperHyperGraph.SomeClassesofSuperHyperGraphandNeutrosophic SuperHyperGrapharecasesofstudy.SomeresultsareappliedinfamilyofSuperHyperGraphandNeutrosophic SuperHyperGraph.

Keywords: SuperHyperGraph;NeutrosophicSuperHyperGraph;Classes;Families

1. Introduction

Fuzzysetin[11],neutrosophicsetin[2],relateddefinitionsofothersetsin[2,8,10],hypergraphsandnewnotionsonthemin[6],neutrosophicgraphsin[3],studiesonneutrosophic graphsin[1],relevantdefinitionsofothergraphsbasedonfuzzygraphsin[7],areproposed. Also,somestudiesandresearchesaboutneutrosophicgraphs,areproposedasabookin[5].

2. SuperHyperGraph

Definition2.1. (Smarandachein2019and2020,[9]).

Anorderedpair(G ⊆ P (V ),E ⊆ P (V ))iscalledby SuperHyperGraph andit’sdenotedby SHG.

HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

Definition2.2. (Smarandachein2019and2020,[9]).

Anorderedpair(Gn ⊆ P n(V ),En ⊆ P n(V ))iscalledby n-SuperHyperGraph andit’s denotedbyn-SHG.

Definition2.3. (Dominating,ResolvingandColoring).

AssumeSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )).

(a): SuperHyper-dominatingsetandnumberaredefinedasfollows.

(i): ASuperVertex Xn SuperHyper-dominates aSuperVertex Yn ifthere’satleast oneSuperHyperEdgewhichhavethem.

(ii): Aset S iscalled SuperHyper-dominatingset ifforevery Yn ∈ Gn \ S, there’s atleastoneSuperVertex Xn whichSuperHyper-dominatesSuperVertex Yn

(iii): If S issetofallsetsofSuperHyper-dominatingsets,then

|X| =min S∈S |{∪Xn|Xn ∈ S}|

iscalled optimal-SuperHyper-dominatingnumber and X iscalled optimalSuperHyper-dominatingset.

(b): SuperHyper-resolvingsetandnumberaredefinedasfollows.

(i): ASuperVertex x SuperHyper-resolves SuperVertices y,w if d(x,y) = d(x,w) .

(ii): Aset S iscalled SuperHyper-resolvingset ifforevery Yn ∈ Gn \ S, there’sat leastoneSuperVertex Xn whichSuperHyper-resolvesSuperVertices Yn,Wn.

(iii): If S issetofallsetsofSuperHyper-resolvingsets,then

|X| =min S∈S |{∪Xn|Xn ∈ S}| iscalled optimal-SuperHyper-resolvingnumber and X iscalled optimalSuperHyper-resolvingset.

(c): SuperHyper-coloringsetandnumberaredefinedasfollows.

(i): ASuperVertex Xn SuperHyper-colors aSuperVertex Yn differentlywithitself ifthere’satleastoneSuperHyperEdgewhichisincidenttothem.

(ii): Aset Sn iscalled SuperHyper-coloringset ifforevery Yn ∈ Gn \ Sn, there’s atleastoneSuperVertex Xn whichSuperHyper-colorsSuperVertex Yn.

(iii): If Sn issetofallsetsofSuperHyper-coloringsets,then |X| =min Sn ∈Sn |{∪Xn|Xn ∈ Sn}| iscalled optimal-SuperHyper-coloringnumber and X iscalled optimalSuperHyper-coloringset.

HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

Proposition2.4. AssumeSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )).S ismaximum setofSuperVerticeswhichformaSuperHyperEdge.Thenoptimal-SuperHyper-coloringsethas ascardinalityas S has.

Proposition2.5. AssumeSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) IfoptimalSuperHyper-coloringnumberis |V |, thenforeverySuperVertexthere’satleastoneSuperHyperEdgewhichcontainshasallmembersof V.

Proposition2.6. AssumeSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) Ifthere’s atleastoneSuperHyperEdgewhichhasallmembersof V, thenoptimal-SuperHyper-coloring numberis |V |.

Proposition2.7. AssumeSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) IfoptimalSuperHyper-dominatingnumberis |V |, thenthere’sonememberof V, iscontainedin,atleast oneSuperVertexwhichdoesn’thaveincidenttoanySuperHyperEdge.

Proposition2.8. AssumeSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )). ThenoptimalSuperHyper-dominatingnumberis < |V |.

Proposition2.9. AssumeSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) IfoptimalSuperHyper-resolvingnumberis |V |, theneverygivenSuperVertexdoesn’thaveincidentto anySuperHyperEdge.

Proposition2.10. AssumeSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )). Then optimal-SuperHyper-resolvingnumberis < |V |.

Proposition2.11. AssumeSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) IfoptimalSuperHyper-coloringnumberis |V |, thenallSuperVerticeswhichhaveincidenttoatleastone SuperHyperEdge.

Proposition2.12. AssumeSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )). Then optimal-SuperHyper-coloringnumberisn’t < |V |.

Proposition2.13. AssumeSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) Then optimal-SuperHyper-dominatingsethascardinalitywhichisgreaterthan n 1 where n is thecardinalityoftheset V.

Proposition2.14. AssumeSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )).S ismaximumsetofSuperVerticeswhichformaSuperHyperEdge.Then S isoptimal-SuperHypercoloringsetand |{∪Xn | Xn ∈ S}| isoptimal-SuperHyper-coloringnumber.

Proposition2.15. AssumeSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )). If S is SuperHyper-dominatingset,then D contains S isSuperHyper-dominatingset. HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

Proposition2.16. AssumeSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )). If S is SuperHyper-resolvingset,then D contains S isSuperHyper-resolvingset.

Proposition2.17. AssumeSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) If S is SuperHyper-coloringset,then D contains S isSuperHyper-coloringset.

Proposition2.18. AssumeSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) Then Gn is SuperHyper-dominatingset.

Proposition2.19. AssumeSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )).Then Gn is SuperHyper-resolvingset.

Proposition2.20. AssumeSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) Then Gn is SuperHyper-coloringset.

Proposition2.21. Assume G isafamilyofSuperHyperGraph.Then Gn isSuperHyperdominatingsetforallmembersof G, simultaneously.

Proposition2.22. Assume G isafamilyofSuperHyperGraph.Then Gn isSuperHyperresolvingsetforallmembersof G, simultaneously.

Proposition2.23. Assume G isafamilyofSuperHyperGraph.Then Gn isSuperHypercoloringsetforallmembersof G, simultaneously.

Proposition2.24. Assume G isafamilyofSuperHyperGraph.Then Gn\{Xn} isSuperHyperdominatingsetforallmembersof G, simultaneously.

Proposition2.25. Assume G isafamilyofSuperHyperGraph.Then Gn\{Xn} isSuperHyperresolvingsetforallmembersof G, simultaneously.

Proposition2.26. Assume G isafamilyofSuperHyperGraph.Then Gn \{Xn} isn’t SuperHyper-coloringsetforallmembersof G, simultaneously.

Proposition2.27. Assume G isafamilyofSuperHyperGraph.ThenunionofSuperHyperdominatingsetsfromeachmemberof G isSuperHyper-dominatingsetforallmembersof G, simultaneously.

Proposition2.28. Assume G isafamilyofSuperHyperGraph.ThenunionofSuperHyperresolvingsetsfromeachmemberof G isSuperHyper-resolvingsetforallmembersof G, simultaneously.

Proposition2.29. Assume G isafamilyofSuperHyperGraph.ThenunionofSuperHypercoloringsetsfromeachmemberof G isSuperHyper-coloringsetforallmembersof G, simultaneously.

HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

Proposition2.30. Assume G isafamilyofSuperHyperGraph.ForeverygivenSuperVertex, there’soneSuperHyperGraphsuchthattheSuperVertexhasanotherSuperVertexwhichare incidenttoaSuperHyperEdge.IfforgivenSuperVertex,allSuperVerticeshaveacommon SuperHyperEdgeinthisway,then Gn \{Xn} isoptimal-SuperHyper-dominatingsetforall membersof G, simultaneously.

Proposition2.31. Assume G isafamilyofSuperHyperGraph.ForeverygivenSuperVertex, there’soneSuperHyperGraphsuchthattheSuperVertexhasanotherSuperVertexwhichare incidenttoaSuperHyperEdge.IfforgivenSuperVertex,allSuperVerticeshaveacommonSuperHyperEdgeinthisway,then Gn \{Xn} isoptimal-SuperHyper-resolvingsetforallmembers of G, simultaneously.

Proposition2.32. Assume G isafamilyofSuperHyperGraph.ForeverygivenSuperVertex, there’soneSuperHyperGraphsuchthattheSuperVertexhasanotherSuperVertexwhichare incidenttoaSuperHyperEdge.IfforgivenSuperVertex,allSuperVerticeshaveacommon SuperHyperEdgeinthisway,then Gn isoptimal-SuperHyper-coloringsetforallmembersof G, simultaneously.

Proposition2.33. Let SHG beaSuperHyperGraph.An (k 1)-setfromank-setoftwin SuperVerticesissubsetofaSuperHyper-resolvingset.

Corollary2.34. Let SHG beaSuperHyperGraph.ThenumberoftwinSuperVerticesis n 1. ThenSuperHyper-resolvingnumberis n 2.

Corollary2.35. Let SHG beSuperHyperGraph.ThenumberoftwinSuperVerticesis n 1 ThenSuperHyper-resolvingnumberis n 2. Every (n 2)-setincludingtwinSuperVerticesis SuperHyper-resolvingset.

Proposition2.36. Let SHG beSuperHyperGraphsuchthatit’scomplete.ThenSuperHyperresolvingnumberis n 1. Every (n 1)-setisSuperHyper-resolvingset.

Proposition2.37. Let G beafamilyofSuperHyperGraphswithcommonsupervertexset Gn. ThensimultaneouslySuperHyper-resolvingnumberof G is |V |− 1

Proposition2.38. Let G beafamilyofSuperHyperGraphswithcommonSuperVertexset Gn. ThensimultaneouslySuperHyper-resolvingnumberof G isgreaterthanthemaximum SuperHyper-resolvingnumberofn-SHG ∈G

Proposition2.39. Let G beafamilyofSuperHyperGraphswithcommonSuperVertexset Gn. ThensimultaneouslySuperHyper-resolvingnumberof G isgreaterthansimultaneously SuperHyper-resolvingnumberof H⊆G HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

Neutrosophic Sets and Systems, Vol. 49, 2022

Theorem2.40. TwinSuperVerticesaren’tSuperHyper-resolvedinanygivenSuperHyperGraph.

Proposition2.41. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beaSuperHyperGraph.IfSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) iscomplete,theneverycoupleofSuperVerticesare twinSuperVertices.

Theorem2.42. Let G beafamilyofSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) with SuperVertexset Gn andn-SHG ∈G iscomplete.ThensimultaneouslySuperHyper-resolving numberis |V |− 1 Every (n 1)-setissimultaneouslySuperHyper-resolvingsetfor G Corollary2.43. Let G beafamilyofSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) with SuperVertexset Gn andn-SHG ∈G iscomplete.ThensimultaneouslySuperHyper-resolving numberis |V |− 1. Every (|V |− 1)-setissimultaneouslySuperHyper-resolvingsetfor G.

Theorem2.44. Let G beafamilyofSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) with SuperVertexset Gn andforeverygivencoupleofSuperVertices,there’san-SHG ∈G such thatinthat,they’retwinSuperVertices.ThensimultaneouslySuperHyper-resolvingnumberis |V |− 1 Every (|V |− 1)-setissimultaneouslySuperHyper-resolvingsetfor G

Theorem2.45. Let G beafamilyofSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) with SuperVertexset Gn. If G containsthreeSuperHyper-starswithdifferentSuperHyper-centers, thensimultaneouslySuperHyper-resolvingnumberis |V |− 2. Every (|V |− 2)-setissimultaneouslySuperHyper-resolvingsetfor G

Corollary2.46. Let G beafamilyofSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) with SuperVertexset Gn If G containsthreeSuperHyper-starswithdifferentSuperHyper-centers, thensimultaneouslySuperHyper-resolvingnumberis |V |− 2. Every (|V |− 2)-setissimultaneouslySuperHyper-resolvingsetfor G

Proposition2.47. ConsidertwoantipodalSuperVertices Xn and Yn inanygiveneven SuperHyper-cycle.Let Un and Vn begivenSuperVertices.Then d(Xn,Un) = d(Xn,Vn) if andonlyif d(Yn,Un) = d(Yn,Vn).

Proposition2.48. ConsidertwoantipodalSuperVertices Xn and Yn inanygivenevencycle. Let Un and Vn begivenSuperVertices.Then d(Xn,Un)= d(Xn,Vn) ifandonlyif d(Yn,Un)= d(Yn,Vn)

Proposition2.49. ThesetcontainstwoantipodalSuperVertices,isn’tSuperHyper-resolving setinanygivenevenSuperHyper-cycle.

HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

Proposition2.50. ConsidertwoantipodalSuperVertices Xn and Yn inanygiveneven SuperHyper-cycle. Xn SuperHyper-resolvesagivencoupleofSuperVertices, Zn and Zn,if andonlyif Yn does.

Proposition2.51. TherearetwoantipodalSuperVerticesaren’tSuperHyper-resolvedbyother twoantipodalSuperVerticesinanygivenevenSuperHyper-cycle.

Proposition2.52. ForanytwoantipodalSuperVerticesinanygivenevenSuperHyper-cycle, thereareonlytwoantipodalSuperVerticesdon’tSuperHyper-resolvethem.

Proposition2.53. InanygivenevenSuperHyper-cycle,foranySuperVertex,there’sonly oneSuperVertexsuchthatthey’reantipodalSuperVertices.

Proposition2.54. LetSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) beaneven SuperHyper-cycle.TheneverycoupleofSuperVerticesareSuperHyper-resolvingsetifand onlyiftheyaren’tantipodalSuperVertices.

Corollary2.55. LetSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) beaneven SuperHyper-cycle.ThenSuperHyper-resolvingnumberistwo.

Corollary2.56. LetSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) beaneven SuperHyper-cycle.ThenSuperHyper-resolvingsetcontainscoupleofSuperVerticessuchthat theyaren’tantipodalSuperVertices.

Corollary2.57. Let G beafamilySuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) be anoddSuperHyper-cyclewithcommonSuperVertexset Gn.ThensimultaneouslySuperHyperresolvingsetcontainscoupleofSuperVerticessuchthattheyaren’tantipodalSuperVerticesand SuperHyper-resolvingnumberistwo.

Proposition2.58. InanygivenSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) whichis oddSuperHyper-cycle,foranySuperVertex,there’snoSuperVertexsuchthatthey’reantipodal SuperVertices.

Proposition2.59. LetSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) beanodd SuperHyper-cycle.TheneverycoupleofSuperVerticesareSuperHyper-resolvingset.

Proposition2.60. LetSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) beanoddcycle. ThenSuperHyper-resolvingnumberistwo.

Corollary2.61. LetSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) beanoddcycle. ThenSuperHyper-resolvingsetcontainscoupleofSuperVertices. HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

Corollary2.62. Let G beafamilyofSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) whichareoddSuperHyper-cycleswithcommonSuperVertexset Gn Thensimultaneously SuperHyper-resolvingsetcontainscoupleofSuperVerticesandSuperHyper-resolvingnumber istwo.

Proposition2.63. LetSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) beaSuperHyperpath.TheneverySuperHyper-leafformsSuperHyper-resolvingset.

Proposition2.64. LetSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) beaSuperHyperpath.ThenasetincludingeverycoupleofSuperVerticesisSuperHyper-resolvingset.

Proposition2.65. LetSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) beaSuperHyperpath.Thenan 1-setcontainsleafisSuperHyper-resolvingsetandSuperHyper-resolvingnumber isone.

Corollary2.66. Let G beafamilyofSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) are SuperHyper-pathswithcommonSuperVertexset Gn suchthatthey’veacommonSuperHyperleaf.ThensimultaneouslySuperHyper-resolvingnumberis 1, 1-setcontainscommonleaf,is simultaneouslySuperHyper-resolvingsetfor G

Proposition2.67. Let G beafamilyofSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) areSuperHyper-pathswithcommonSuperVertexset Gn suchthatforeverySuperHyper-leaf Ln fromn-SHG, there’sanothern-SHG ∈G suchthat Ln isn’tSuperHyper-leaf.Thenan 2-set containseverycoupleofSuperVertices,isSuperHyper-resolvingset.An 2-setcontainsevery coupleofSuperVertices,isoptimal-SuperHyper-resolvingset.Optimal-SuperHyper-resolving numberistwo.

Corollary2.68. Let G beafamilyofSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) are SuperHyper-pathswithcommonSuperVertexset Gn suchthatthey’venocommonSuperHyperleaf.Thenan 2-setissimultaneouslyoptimal-SuperHyper-resolvingsetandsimultaneously optimal-SuperHyper-resolvingnumberis 2.

Proposition2.69. LetSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) beaSuperHyper-tpartite.TheneverysetexcludingcoupleofSuperVerticesindifferentpartswhosecardinalities ofthemarestrictlygreaterthanone,isoptimal-SuperHyper-resolvingset.

Corollary2.70. LetSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) beaSuperHyper-tpartite.Let |V |≥ 3 Thenevery (|V |− 2)-setexcludestwoSuperVerticesfromdifferentparts whosecardinalitiesofthemarestrictlygreaterthanone,isoptimal-SuperHyper-resolvingset andoptimal-SuperHyper-resolvingnumberis |V |− 2 HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

Corollary2.71. LetSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) beaSuperHyperbipartite.Let |V |≥ 3 Thenevery (|V |−2)-setexcludestwoSuperVerticesfromdifferentparts, isoptimal-SuperHyper-resolvingsetandoptimal-SuperHyper-resolvingnumberis |V |− 2 Corollary2.72. LetSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) beaSuperHyperstar.Thenevery (|V |−2)-setexcludesSuperHyper-centerandagivenSuperVertex,isoptimalSuperHyper-resolvingsetandoptimal-SuperHyper-resolvingnumberis (|V |− 2)

Corollary2.73. LetSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) beaSuperHyperwheel.Let |V |≥ 3 Thenevery (|V |− 2)-setexcludesSuperHyper-centerandagivenSuperVertex,isoptimal-SuperHyper-resolvingsetandoptimal-SuperHyper-resolvingnumberis |V |− 2.

Corollary2.74. Let G beafamilyofSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) whichareSuperHyper-t-partitewithcommonSuperVertexset Gn Let |V |≥ 3 Thensimultaneouslyoptimal-SuperHyper-resolvingnumberis |V |− 2 andevery (|V |− 2)-setexcludestwo SuperVerticesfromdifferentparts,issimultaneouslyoptimal-SuperHyper-resolvingsetfor G. Corollary2.75. Let G beafamilyofSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) whichareSuperHyper-bipartitewithcommonSuperVertexset Gn. Let |V |≥ 3. Thensimultaneouslyoptimal-SuperHyper-resolvingnumberis |V |− 2 andevery (|V |− 2)-setexcludestwo SuperVerticesfromdifferentparts,issimultaneouslyoptimal-SuperHyper-resolvingsetfor G. Corollary2.76. Let G beafamilyofSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) whichareSuperHyper-starwithcommonSuperVertexset Gn. Let |V |≥ 3. Thensimultaneously optimal-SuperHyper-resolvingnumberis |V |− 2 andevery (|V |− 2)-setexcludesSuperHypercenterandagivenSuperVertex,issimultaneouslyoptimal-SuperHyper-resolvingsetfor G Corollary2.77. Let G beafamilyofSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) whichareSuperHyper-wheelwithcommonSuperVertexset Gn. Let |V |≥ 3. Thensimultaneouslyoptimal-SuperHyper-resolvingnumberis |V |− 2 andevery (|V |− 2)-setexcludes SuperHyper-centerandagivenSuperVertex,issimultaneouslyoptimal-SuperHyper-resolving setfor G.

Proposition2.78. LetSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) beaSuperHypercomplete.Thenoptimal-SuperHyper-coloringnumberis |V |.

Proposition2.79. LetSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) beaSuperHyperpath.Thenoptimal-SuperHyper-coloringnumberistwo.

Proposition2.80. LetSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) beaneven SuperHyper-cycle.Thenoptimal-SuperHyper-coloringnumberistwo. HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

Proposition2.81. LetSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) beanodd SuperHyper-cycle.Thenoptimal-SuperHyper-coloringnumberisthree.

Proposition2.82. LetSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) beaSuperHyperstar.Thenoptimal-SuperHyper-coloringnumberistwo.

Proposition2.83. LetSuperHyperGraphs SHG =(G ⊆ P (V ),E ⊆ P (V )) beaSuperHyperwheelsuchthatithasevenSuperHyper-cycle.Thenoptimal-SuperHyper-coloringnumberis Three.

Proposition2.84. LetSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) beaSuperHyperwheelsuchthatithasoddSuperHyper-cycle.Thenoptimal-SuperHyper-coloringnumberis four.

Proposition2.85. LetSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) beaSuperHypercompleteandSuperHyper-bipartite.Thenoptimal-SuperHyper-coloringnumberistwo.

Proposition2.86. LetSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) beaSuperHypercompleteandSuperHyper-t-partite.Thenoptimal-SuperHyper-coloringnumberis t.

Proposition2.87. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beSuperHyperGraph.ThenoptimalSuperHyper-coloringnumberis 1 ifandonlyif SHG =(G ⊆ P (V ),E ⊆ P (V )) isSuperHyperempty.

Proposition2.88. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beSuperHyperGraph.ThenoptimalSuperHyper-coloringnumberis 2 ifandonlyif SHG =(G ⊆ P (V ),E ⊆ P (V )) isboth SuperHyper-completeandSuperHyper-bipartite.

Proposition2.89. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beSuperHyperGraph.Then optimal-SuperHyper-coloringnumberis |V | ifandonlyif SHG =(G ⊆ P (V ),E ⊆ P (V )) isSuperHyper-complete.

Proposition2.90. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beSuperHyperGraph.ThenoptimalSuperHyper-coloringnumberisobtainedfromthenumberofSuperVerticeswhichis |Gn| and optimal-SuperHyper-coloringnumberisatmost |V |

Proposition2.91. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beSuperHyperGraph.ThenoptimalSuperHyper-coloringnumberisatmost ∆+1 andatleast 2.

Proposition2.92. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beSuperHyperGraphand SuperHyper-r-regular.Thenoptimal-SuperHyper-coloringnumberisatmost r +1.

Definition2.93. (Eulerian(Hamiltonian)NeutrosophicPath).

Let SHG =(G ⊆ P (V ),E ⊆ P (V ))beaneutrosophicSuperHyperGraph.Then HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

(i) Eulerian(Hamiltonian)neutrosophicpath Me(SHG)(Mh(SHG))foraneutrosophicSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V ))isasequenceofconsecutive edges(vertices) x1,x2, ,xS(SHG)(x1,x2, ,xO(SHG))whichisneutrosophicpath; (ii) n-Eulerian(Hamiltonian)neutrosophicpath Ne(SHG)(Nh(SHG))foraneutrosophicSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V ))isthenumberofsequences ofconsecutiveedges(vertices) x1,x2, ,xS(SHG)(x1,x2, ,xO(SHG))whichisneutrosophicpath.

Proposition2.94. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-neutrosophicSuperHyperGraphwithtwoweakestedges.Then

Me(CMT σ): NotExisted; Mh(CMT σ): vτ (1),vτ (2), ··· ,vτ (O(CMT σ ) 1),vτ (O(CMT σ )) where τ isapermutationon O(CMT σ). Ne(CMT σ)=0; Nh(CMT σ)= O(CMT σ)!

Proposition2.95. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beapath-neutrosophicSuperHyperGraph.Then

Me(PTH): v1,v2, ,vS(PTH); Mh(PTH): v1,v2, ,vO(PTH)

Ne(PTH)=1; Nh(PTH)=1

Proposition2.96. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacycle-neutrosophicSuperHyperGraphwhere O(CYC) ≥ 3 Then

Me(CYC): NotExisted;

Mh(CYC): xi,xi+1, ,xO(CYC) 1,xO(CYC), ,xi 1.

Ne(CYC)=0; Nh(CYC)= O(CYC) HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

Proposition2.97. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beastar-neutrosophicSuperHyperGraphwithcenter c. Then

Me(STR1,σ2 ): v1,v2

Mh(STR1,σ2 ): v1,c,v2

where O(STR1,σ2 ) ≤ 2;

Me(STR1,σ2 ): NotExisted

Mh(STR1,σ2 ): NotExisted where O(STR1,σ2 ) ≥ 3.

Ne(STR1,σ2 )=2

Nh(STR1,σ2 )=3 where O(STR1,σ2 ) ≤ 2;

Ne(STR1,σ2 )=0

Nh(STR1,σ2 )=0

where O(STR1,σ2 ) ≥ 3

Proposition2.98. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-bipartite-neutrosophic SuperHyperGraph.Then

Me(CMCσ1,σ2 ): NotExisted

Mh(CMCσ1,σ2 ): v1,v2, ··· ,vO(CMCσ1 ,σ2 ) 1,vO(CMCσ1 ,σ2 ) where O(CMCσ1,σ2 ) ≥ 3, |V1| = |V2|,v2i+1 ∈ V1,v2i ∈ V2;

Me(CMCσ1,σ2 ): v1v2

Mh(CMCσ1,σ2 ): v1,v2 where O(CMCσ1,σ2 )=2;

Me(CMCσ1,σ2 ): Mh(CMCσ1,σ2 ): v1 where O(CMCσ1,σ2 )=1.

Ne(CMCσ1,σ2 )=0

Nh(CMCσ1,σ2 )= c where O(CMCσ1,σ2 ) ≥ 3, |V1| = |V2|,v2i+1 ∈ V1,v2i ∈ V2;

Ne(CMCσ1,σ2 )=2

Nh(CMCσ1,σ2 )=2 where O(CMCσ1,σ2 )=2; Ne(CMCσ1,σ2 )=

HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

Nh(CMCσ1,σ2 )=1 where O(CMCσ1,σ2 )=1

Proposition2.99. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-t-partite-neutrosophic SuperHyperGraph.Then

Me(CMCσ1,σ2, ,σt ): NotExisted

Mh(CMCσ1,σ2, ,σt ): v1,v2, ,vO(CMCσ1 ,σ2 , ,σt ) 1,vO(CMCσ1 ,σ2 , ,σt ) where O(CMCσ1,σ2, ,σt ) ≥ 3, |Vi| = |Vj |,v2i+1 ∈ Vi,v2i ∈ Vj ;

Me(CMCσ1,σ2, ,σt ): v1v2

Mh(CMCσ1,σ2, ,σt ): v1,v2 where O(CMCσ1,σ2, ,σt )=2;

Me(CMCσ1,σ2, ··· ,σt ):

Mh(CMCσ1,σ2, ,σt ): v1 where O(CMCσ1,σ2, ,σt )=1

Ne(CMCσ1,σ2, ,σt )=0

Nh(CMCσ1,σ2, ,σt )= c where O(CMCσ1,σ2, ,σt ) ≥ 3, |Vi| = |Vj |,v2i+1 ∈ Vi,v2i ∈ Vj ;

Ne(CMCσ1,σ2, ,σt )=2

Nh(CMCσ1,σ2, ,σt )=2 where O(CMCσ1,σ2, ,σt )=2;

Ne(CMCσ1,σ2, ,σt )= Nh(CMCσ1,σ2, ,σt )=1 where O(CMCσ1,σ2, ,σt )=1.

Proposition2.100. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beawheel-neutrosophicSuperHyperGraph.Then

Mh(WHL1,σ2 ): xi,xi+1, ··· ,xO(WHL1,σ2 ) 1,xO(WHL1,σ2 ),xi 1

Me(WHL1,σ2 ): v1,v2,v3 where S(WHL1,σ2 )=3

Mh(WHL1,σ2 ): xi,xi+1, ,xO(WHL1,σ2 ) 1,xO(WHL1,σ2 ),xi 1

Me(WHL1,σ2 ): NotExisted where S(WHL1,σ2 ) > 3. Nh(WHL1,σ2 )= O(WHL1,σ2 );

HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

Ne(WHL1,σ2 )=3; where S(WHL1,σ2 )=3.

Nh(WHL1,σ2 )= O(WHL1,σ2 );

Ne(WHL1,σ2 )=0; where S(WHL1,σ2 ) > 3.

3. NeutrosophicSuperHyperGraph

Definition3.1. (ZeroForcingNumber).

Let SHG =(G ⊆ P (V ),E ⊆ P (V ))beaneutrosophicSuperHyperGraph.Then

(i) zeroforcingnumber Z(SHG)foraneutrosophicSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V ))isminimumcardinalityofaset S ofblackvertices(whereasvertices in V (G) \ S arecoloredwhite)suchthat V (G)isturnedblackafterfinitelymany applicationsof“thecolor-changerule”:awhitevertexisconvertedtoablackvertex ifitistheonlywhiteneighborofablackvertex;

(ii) zeroforcingneutrosophic-number Zn(SHG)foraneutrosophicSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V ))isminimumneutrosophiccardinalityofa set S ofblackvertices(whereasverticesin V (G) \ S arecoloredwhite)suchthat V (G) isturnedblackafterfinitelymanyapplicationsof“thecolor-changerule”:awhite vertexisconvertedtoablackvertexifitistheonlywhiteneighborofablackvertex.

Definition3.2. (IndependentNumber).

Let SHG =(G ⊆ P (V ),E ⊆ P (V ))beaneutrosophicSuperHyperGraph.Then

(i) independentnumber I(SHG)foraneutrosophicSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V ))ismaximumcardinalityofaset S ofverticessuchthateverytwo verticesof S aren’tendpointsforanedge,simultaneously;

(ii) independentneutrosophic-number In(SHG)foraneutrosophicSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V ))ismaximumneutrosophiccardinalityofa set S ofverticessuchthateverytwoverticesof S aren’tendpointsforanedge,simultaneously.

Definition3.3. (CliqueNumber).

Let SHG =(G ⊆ P (V ),E ⊆ P (V ))beaneutrosophicSuperHyperGraph.Then

(i) cliquenumber C(SHG)foraneutrosophicSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V ))ismaximumcardinalityofaset S ofverticessuchthateverytwo verticesof S areendpointsforanedge,simultaneously; HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

(ii) cliqueneutrosophic-number Cn(SHG)foraneutrosophicSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V ))ismaximumneutrosophiccardinalityofaset S ofverticessuchthateverytwoverticesof S areendpointsforanedge,simultaneously.

Definition3.4. (MatchingNumber).

Let SHG =(G ⊆ P (V ),E ⊆ P (V ))beaneutrosophicSuperHyperGraph.Then

(i) matchingnumber M(SHG)foraneutrosophicSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V ))ismaximumcardinalityofaset S ofedgessuchthateverytwo edgesof S don’thaveanyvertexincommon;

(ii) matchingneutrosophic-number Mn(SHG)foraneutrosophicSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V ))ismaximumneutrosophiccardinalityofaset S of edgessuchthateverytwoedgesof S don’thaveanyvertexincommon.

Definition3.5. (GirthandNeutrosophicGirth).

Let SHG =(G ⊆ P (V ),E ⊆ P (V ))beaneutrosophicSuperHyperGraph.Then (i) girth G(SHG)foraneutrosophicSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V )) isminimumcrispcardinalityofverticesformingshortestcycle.Ifthereisn’t,then girthis ∞; (ii) neutrosophicgirth Gn(SHG)foraneutrosophicSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V ))isminimumneutrosophiccardinalityofverticesformingshortest cycle.Ifthereisn’t,thengirthis ∞

Proposition3.6. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-neutrosophicSuperHyperGraph.Then (1) Z(CMT σ)= O(CMT σ) 1. (2) I(SHG)=1. (3) C(SHG)= O(SHG) (4) M(SHG)= n 2 (5) G(SHG)=3.

Proposition3.7. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beapath-neutrosophicSuperHyperGraph.Then

HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

(1)

Z(PTH n)=1. (2)

I(SHG)= O(SHG) 2 (3) C(SHG)=2. (4) M(SHG)= n 2 (5) G(SHG)= ∞

Proposition3.8. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacycle-neutrosophicSuperHyperGraphwhere O(CYC) ≥ 3 Then (1) Z(CYCn)=2 (2)

I(SHG)= O(SHG) 2 . (3) C(SHG)=2. (4) M(SHG)= n 2 (5) G(SHG)= O(SHG).

Proposition3.9. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beastar-neutrosophicSuperHyperGraphwithcenter c. Then (1)

Z(STR1,σ2 )= O(STR1,σ2 ) 2 (2)

I(SHG)= O(SHG) 1. (3) C(SHG)=2. (4) M(SHG)=1

HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

(5) G(SHG)= ∞.

Proposition3.10. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-bipartite-neutrosophic SuperHyperGraph.Then (1)

Z(CMT σ1,σ2 )= O(CMT σ1,σ2 ) 2 (2)

I(SHG)=max{|V1|, |V2|} (3) C(SHG)=2 (4) M(SHG)=min{|V1|, |V2|} (5) G(SHG)=4 where O(SHG) ≥ 4 And G(SHG)= ∞ where O(SHG) ≤ 3

Proposition3.11. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-t-partite-neutrosophic SuperHyperGraph.Then (1)

Z(CMT σ1,σ2, ,σt )= O(CMT σ1,σ2, ,σt ) 1. (2)

I(SHG)=max{|V1|, |V2|, ··· , |Vt|}. (3) C(SHG)= t. (4)

M(SHG)=min |Vi|t i=1 (5) G(SHG)=3 where t ≥ 3 G(SHG)=4 where t ≤ 2 And G(SHG)= ∞

HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

where O(SHG) ≤ 2.

Proposition3.12. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-neutrosophicSuperHyperGraph.Then (1)

Zn(CMT σ)= On(CMT σ) max{Σ3 i=1σi(x)}x∈V . (2) In(SHG)=max{ 3 i=1 σi(x)}x∈V . (3) Cn(SHG)= On(SHG). (4)

Mn(SHG)=max{ 3 i=1 µi(x0x1)+ 3 i=1 µi(x1x2)+ ··· + 3 i=1 µi(xj 1xj )}j= n 2 (5) Gn(SHG)=min{Σ3 i=1(σi(x)+ σi(y)+ σi(z))}

Proposition3.13. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beapath-neutrosophicSuperHyperGraph.Then (1) Zn(PTH n)=min{Σ3 i=1σi(x)}x isaleaf . (2)

In(SHG)=max{ 3 i=1 (σi(x1)+ σi(x3)+ + σi(xt)), 3 i=1 σi(x2)+ σi(x4)+ + σi(xt))}xi xi+1∈E (3) Cn(SHG)=max{ 3 i=1 (σi(xj )+ σi(xj+1))}xj xj+1∈E (4)

Mn(SHG)=max{ 3 i=1 µi(x0x1)+ 3 i=1 µi(x2x3)+ + 3 i=1 µi(xj 1xj )}|S|= n 2 . (5) Gn(SHG)= ∞

Proposition3.14. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacycle-neutrosophicSuperHyperGraphwhere O(CYC) ≥ 3 Then

HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

(1)

Zn(CYCn)=min{Σ3 i=1σi(x)+Σ3 i=1σi(y)}xy∈E.. (2)

In(SHG)=max{ 3 i=1 (σi(x1)+ σi(x3)+ + σi(xt)), 3 i=1 σi(x2)+ σi(x4)+ ··· + σi(xt))}xi xi+1∈E . (3) Cn(SHG)=max{ 3 i=1 (σi(xj )+ σi(xj+1))}xj xj+1∈E (4) Mn(SHG)=max{ 3 i=1 µi(x0x1)+ 3 i=1 µi(x2x3)+ ··· + 3 i=1 µi(xj 1xj )}|S|= n 2 . (5) Gn(SHG)= On(SHG)

Proposition3.15. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beastar-neutrosophicSuperHyperGraphwithcenter c. Then (1) Zn(STR1,σ2 )= On(STR1,σ2 ) max{Σ3 i=1σi(c)+Σ3 i=1σi(x)}x∈V . (2) In(SHG)= On(SHG) σ(c)= 3 i=1 xj =c

σi(xj ). (3) Cn(SHG)= 3 i=1 σi(c)+max{ 3 i=1 σi(xj )}. (4)

Mn(SHG)=max{ 3 i=1 µi(xj 1xj )}xj 1xj ∈E . (5) Gn(SHG)= ∞

Proposition3.16. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-bipartite-neutrosophic SuperHyperGraph.Then (1)

Zn(CMT σ1,σ2 )= On(CMT σ1,σ2 ) max{Σ3 i=1σi(x)+Σ3 i=1σi(x )}x,x ∈V HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

Sets and Systems, Vol. 49, 2022 550

(2)

In(SHG)=max{( 3 i=1 xj ∈V1

σi(xj )), ( 3 i=1 xj ∈V2

σi(xj ))}

(3) Cn(SHG)=max{ 3 i=1 (σi(xj )+ σi(xj ))}xj ∈V1,xj ∈V2 (4)

Mn(SHG)=max{ 3 i=1 µi(x0x1)+ 3 i=1 µi(x2x3)+ + 3 i=1 µi(xj 1xj )}|S|=min{|V1|,|V2|}.

(5) Gn(SHG)=min{Σ3 i=1(σi(x)+ σi(y)+ σi(z)+ σi(w))}x,y∈V1,z,w∈V2 where O(SHG) ≥ 4 and min{|V1|, |V2|}≥ 2 Also, Gn(SHG)= ∞ where O(SHG) ≤ 3

Proposition3.17. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-t-partite-neutrosophic SuperHyperGraph.Then

(1) Zn(CMT σ1,σ2, ,σt )= On(CMT σ1,σ2, ,σt ) max{Σ3 i=1σi(x)}x∈V (2)

σi(xj )), ( 3 i=1 xj ∈V2

σi(xj )), , ( 3 i=1 xj ∈Vt Neutrosophic

σi(xj ))} (3) Cn(SHG)=max{ 3 i=1 (σi(xj1 )+ σi(xj2 )+ ··· + σi(xjt ))}xj1 ∈V1,xj2 ∈V2, ,xjt ∈Vt . (4) Mn(SHG)=max{ 3 i=1 µi(x0x1)+ 3 i=1 µi(x2x3)+ ··· + 3 i=1 µi(xj 1xj )}|S|=min |Vi |t i=1}. (5) Gn(SHG)=min{Σ3 i=1(σi(x)+ σi(y)+ σi(z))}x∈V1,y∈V2,z∈V3 where t ≥ 3 Gn(SHG)=min{Σ3 i=1(σi(x)+ σi(y)+ σi(z)+ σi(w))}x,y∈V1,z,w∈V2 HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

Systems, Vol. 49, 2022

where t ≤ 2. And Gn(SHG)= ∞ where O(SHG) ≤ 2

3.1. SettingofNeutrosophic1-Zero-ForcingNumber

Definition3.18. (1-Zero-ForcingNumber).

Let SHG =(G ⊆ P (V ),E ⊆ P (V ))beaneutrosophicSuperHyperGraph.Then

(i) 1-zero-forcingnumber Z(SHG)foraneutrosophicSuperHyperGraph SHG = (G ⊆ P (V ),E ⊆ P (V ))isminimumcardinalityofaset S ofblackvertices(whereas verticesin V (G) \ S arecoloredwhite)suchthat V (G)isturnedblackafterfinitely manyapplicationsof“thecolor-changerule”:awhitevertexisconvertedtoablack vertexifitistheonlywhiteneighborofablackvertex.Thelastconditionisasfollows. Foronetime,blackcanchangeanyvertexfromwhitetoblack.

(ii) 1-zero-forcingneutrosophic-number Zn(SHG)foraneutrosophicSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V ))isminimumneutrosophiccardinalityofaset S ofblackvertices(whereasverticesin V (G) \ S arecoloredwhite)suchthat V (G)is turnedblackafterfinitelymanyapplicationsof“thecolor-changerule”:awhitevertex isconvertedtoablackvertexifitistheonlywhiteneighborofablackvertex.The lastconditionisasfollows.Foronetime,blackcanchangeanyvertexfromwhiteto black.

Definition3.19. (Failed1-Zero-ForcingNumber).

Let SHG =(G ⊆ P (V ),E ⊆ P (V ))beaneutrosophicSuperHyperGraph.Then

(i) failed1-zero-forcingnumber Z (SHG)foraneutrosophicSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V ))ismaximumcardinalityofaset S ofblackvertices(whereasverticesin V (G) \ S arecoloredwhite)suchthat V (G)isn’tturned blackafterfinitelymanyapplicationsof“thecolor-changerule”:awhitevertexis convertedtoablackvertexifitistheonlywhiteneighborofablackvertex.Thelast conditionisasfollows.Foronetime,Blackcanchangeanyvertexfromwhitetoblack. Thelastconditionisasfollows.Foronetime,blackcanchangeanyvertexfromwhite toblack;

(ii) failed1-zero-forcingneutrosophic-number Zn(SHG)foraneutrosophicSuperHyperGraph SHG =(G ⊆ P (V ),E ⊆ P (V ))ismaximumneutrosophiccardinalityof aset S ofblackvertices(whereasverticesin V (G) \ S arecoloredwhite)suchthat V (G)isn’tturnedblackafterfinitelymanyapplicationsof“thecolor-changerule”:a whitevertexisconvertedtoablackvertexifitistheonlywhiteneighborofablack

HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

vertex.Thelastconditionisasfollows.Foronetime,Blackcanchangeanyvertex fromwhitetoblack.Thelastconditionisasfollows.Foronetime,blackcanchange anyvertexfromwhitetoblack.

Proposition3.20. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-neutrosophicSuperHyperGraph.Then

Z(CMT σ)= O(CMT σ) 2.

Proposition3.21. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beapath-neutrosophicSuperHyperGraph.Then

Z(PTH n)=1

Proposition3.22. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacycle-neutrosophicSuperHyperGraphwhere O(CYC) ≥ 3 Then

Z(CYCn)=1.

Proposition3.23. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beastar-neutrosophicSuperHyperGraphwithcenter c. Then

Z(STR1,σ2 )= O(STR1,σ2 ) 3.

Proposition3.24. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-bipartite-neutrosophic SuperHyperGraph.Then

Z(CMT σ1,σ2 )= O(CMT σ1,σ2 ) 3

Proposition3.25. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-t-partite-neutrosophic SuperHyperGraph.Then

Z(CMT σ1,σ2, ,σt )= O(CMT σ1,σ2, ,σt ) 2

3.2. Settingof1-Zero-ForcingNeutrosophic-Number

Proposition3.26. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-neutrosophicSuperHyperGraph.Then

Zn(CMT σ)= On(CMT σ) max{Σ3 i=1σi(x)+Σ3 i=1σi(y)}x,y∈V .

Proposition3.27. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beapath-neutrosophicSuperHyperGraph.Then

Zn(PTH n)=min{Σ3 i=1σi(x)}x isavertex

Proposition3.28. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacycle-neutrosophicSuperHyperGraphwhere O(CYC) ≥ 3 Then

Zn(CYCn)=min{Σ3 i=1σi(x)}x isavertex

HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

Proposition3.29. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beastar-neutrosophicSuperHyperGraphwithcenter c. Then

Zn(STR1,σ2 )= On(STR1,σ2 ) max{Σ3 i=1σi(c)+Σ3 i=1σi(x)+Σ3 i=1σi(y)}x,y∈V .

Proposition3.30. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-bipartite-neutrosophic SuperHyperGraph.Then

Zn(CMT σ1,σ2 )= On(CMT σ1,σ2 ) max{Σ3 i=1σi(x)+Σ3 i=1σi(x )+Σ3 i=1σi(x )}x,x ,x ∈V

Proposition3.31. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-t-partite-neutrosophic SuperHyperGraph.Then

Zn(CMT σ1,σ2, ,σt )= On(CMT σ1,σ2, ,σt ) max{Σ3 i=1σi(x)+Σ3 i=1σi(x )}x,x ∈V .

3.3. SettingofNeutrosophicFailed1-Zero-ForcingNumber

Proposition3.32. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-neutrosophicSuperHyperGraph.Then Z (CMT σ)= O(CMT σ) 3

Proposition3.33. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beapath-neutrosophicSuperHyperGraph.Then

Z (PTH n)=0

Proposition3.34. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacycle-neutrosophicSuperHyperGraphwhere O(CYC) ≥ 3

Z (CYCn)=0

Proposition3.35. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beastar-neutrosophicSuperHyperGraphwithcenter c. Then

Z (STR1,σ2 )= O(STR1,σ2 ) 4

Proposition3.36. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-bipartite-neutrosophic SuperHyperGraph.Then

Z (CMT σ1,σ2 )= O(CMT σ1,σ2 ) 4.

Proposition3.37. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-t-partite-neutrosophic SuperHyperGraph.Then

Z (CMT σ1,σ2, ,σt )= O(CMT σ1,σ2, ,σt ) 3

HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

3.4. SettingofFailed1-Zero-ForcingNeutrosophic-Number

Proposition3.38. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-neutrosophicSuperHyperGraph.Then

Zn(CMT σ)= On(CMT σ) min{Σ3 i=1σi(x)+Σ3 i=1σi(y)+Σ3 i=1σi(z)}x,y,z∈V .

Proposition3.39. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beapath-neutrosophicSuperHyperGraph.Then

Zn(PTH n)=0

Proposition3.40. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacycle-neutrosophicSuperHyperGraphwhere O(CYC) ≥ 3. Then

Zn(CYCn)=0

Proposition3.41. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beastar-neutrosophicSuperHyperGraphwithcenter c. Then

Zn(STR1,σ2 )= On(STR1,σ2 ) min{Σ3 i=1σi(c)+Σ3 i=1σi(x)+Σ3 i=1σi(y)+Σ3 i=1σi(z)}x,y,z∈V .

Proposition3.42. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-bipartite-neutrosophic SuperHyperGraph.Then Zn(CMT σ1,σ2 )= On(CMT σ1,σ2 ) min{Σ3 i=1σi(x)+Σ3 i=1σi(x )+Σ3 i=1σi(x )+Σ3 i=1σi(x )}x,x ,x ,x ∈V .

Proposition3.43. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beacomplete-t-partite-neutrosophic SuperHyperGraph.Then Zn(CMT σ1,σ2, ,σt )= On(CMT σ1,σ2, ,σt ) min{Σ3 i=1σi(x)+Σ3 i=1σi(x )+Σ3 i=1σi(x )}x,x ∈V .

3.5. GlobalOffensiveAlliance

Definition3.44. Let SHG =(G ⊆ P (V ),E ⊆ P (V ))beaneutrosophicSuperHyperGraph. Then (i) aset S iscalled global-offensivealliance if ∀a ∈ V \ S, |Ns(a) ∩ S| > |Ns(a) ∩ (V \ S)|; (ii) ∀S ⊆ S,S isglobaloffensivealliancebut S isn’tglobaloffensivealliance.Then S is called minimal-global-offensivealliance; HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

(iii) minimal-global-offensive-alliancenumber of SHG is

S isaminimal-global-offensivealliance. |S| andit’sdenotedbyΓ;

(iv) minimal-global-offensive-alliance-neutrosophicnumber of SHG is

Σs∈S Σ3 i=1σi(s) andit’sdenotedbyΓs

S isaminimal-global-offensivealliance.

Proposition3.45. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beastrongneutrosophicSuperHyperGraph.If S isglobal-offensivealliance,then ∀v ∈ V \ S, ∃x ∈ S suchthat (i) v ∈ Ns(x); (ii) vx ∈ E.

Definition3.46. Let SHG =(G ⊆ P (V ),E ⊆ P (V ))beastrongneutrosophicSuperHyperGraph.Suppose S isasetofvertices.Then

(i) S iscalled dominatingset if ∀v ∈ V \S, ∃s ∈ S suchthateither v ∈ Ns(s)or vs ∈ E; (ii) |S| iscalled chromaticnumber if ∀v ∈ V, ∃s ∈ S suchthateither v ∈ Ns(s)or vs ∈ E implies s and v havedifferentcolors.

Proposition3.47. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beastrongneutrosophicSuperHyperGraph.If S isglobal-offensivealliance,then

(i) S isdominatingset; (ii) there’s S ⊆ S suchthat |S | ischromaticnumber.

Proposition3.48. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beastrongneutrosophicSuperHyperGraph.Then

(i) Γ ≤O; (ii) Γs ≤On.

Proposition3.49. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beastrongneutrosophicSuperHyperGraphwhichisconnected.Then

(i) Γ ≤O− 1; (ii) Γs ≤On Σ3 i=1σi(x)

Proposition3.50. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beanoddpath.Then

(i) theset S = {v2,v4, ,vn 1} isminimal-global-offensivealliance; (ii) Γ= n 2 +1 andcorrespondedsetis S = {v2,v4, ··· ,vn 1}; (iii) Γs =min{Σs∈S={v2,v4, ,vn 1}Σ3 i=1σi(s), Σs∈S={v1,v3, ,vn 1}Σ3 i=1σi(s)};

HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

(iv) thesets S1 = {v2,v4, ,vn 1} and S2 = {v1,v3, ,vn 1} areonlyminimal-globaloffensivealliances.

Proposition3.51. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beanevenpath.Then (i) theset S = {v2,v4, .vn} isminimal-global-offensivealliance; (ii) Γ= n 2 andcorrespondedsetsare {v2,v4, ··· .vn} and {v1,v3, ··· .vn 1}; (iii) Γs =min{Σs∈S={v2,v4, ,vn }Σ3 i=1σi(s), Σs∈S={v1,v3, .vn 1}Σ3 i=1σi(s)}; (iv) thesets S1 = {v2,v4, .vn} and S2 = {v1,v3, .vn 1} areonlyminimal-globaloffensivealliances.

Proposition3.52. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beanevencycle.Then (i) theset S = {v2,v4, ,vn} isminimal-global-offensivealliance; (ii) Γ= n 2 andcorrespondedsetsare {v2,v4, ,vn} and {v1,v3, ,vn 1}; (iii) Γs =min{Σs∈S={v2,v4, ,vn }σ(s), Σs∈S={v1,v3, ,vn 1}σ(s)}; (iv) thesets S1 = {v2,v4, ,vn} and S2 = {v1,v3, ,vn 1} areonlyminimal-globaloffensivealliances.

Proposition3.53. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beanoddcycle.Then (i) theset S = {v2,v4, ··· ,vn 1} isminimal-global-offensivealliance; (ii) Γ= n 2 +1 andcorrespondedsetis S = {v2,v4, ,vn 1}; (iii) Γs =min{Σs∈S={v2,v4, .vn 1}Σ3 i=1σi(s), Σs∈S={v1,v3, .vn 1}Σ3 i=1σi(s)}; (iv) thesets S1 = {v2,v4, .vn 1} and S2 = {v1,v3, .vn 1} areonlyminimal-globaloffensivealliances.

Proposition3.54. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) bestar.Then

(i) theset S = {c} isminimal-global-offensivealliance; (ii) Γ=1; (iii) Γs =Σ3 i=1σi(c); (iv) thesets S = {c} and S ⊂ S areonlyglobal-offensivealliances.

Proposition3.55. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) bewheel.Then

(i) theset S = {v1,v3}∪{v6,v9 ,vi+6, ,vn}6+3(i 1)≤n i=1 isminimal-global-offensive alliance; (ii) Γ= |{v1,v3}∪{v6,v9 ··· ,vi+6, ··· ,vn}6+3(i 1)≤n i=1 |; (iii) Γs =Σ{v1,v3}∪{v6,v9 ,vi+6, ,vn }6+3(i 1)≤n i=1 Σ3 i=1σi(s); (iv) theset {v1,v3}∪{v6,v9 ··· ,vi+6, ··· ,vn}6+3(i 1)≤n i=1 isonlyminimal-global-offensive alliance.

Proposition3.56. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beanoddcomplete.Then

HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

(i) theset S = {vi} n 2 +1 i=1 isminimal-global-offensivealliance;

(ii) Γ= n 2 +1;

(iii) Γs =min{Σs∈S Σ3 i=1σi(s)}S={vi } n 2 +1 i=1 ;

(iv) theset S = {vi} n 2 +1 i=1 isonlyminimal-global-offensivealliances.

Proposition3.57. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beanevencomplete.Then

(i) theset S = {vi} n 2 i=1 isminimal-global-offensivealliance; (ii) Γ= n 2 ; (iii) Γs =min{Σs∈S Σ3 i=1σi(s)}S={vi } n 2 i=1 ;

(iv) theset S = {vi} n 2 i=1 isonlyminimal-global-offensivealliances.

Proposition3.58. Let G bea m-familyofneutrosophicstarswithcommonneutrosophic vertexset.Then

(i) theset S = {c1,c2, ··· ,cm} isminimal-global-offensivealliancefor G; (ii) Γ= m for G; (iii) Γs =Σm i=1Σ3 j=1σj (ci) for G;

(iv) thesets S = {c1,c2, ··· ,cm} and S ⊂ S areonlyminimal-global-offensivealliances for G

Proposition3.59. Let G bea m-familyofoddcompletegraphswithcommonneutrosophic vertexset.Then

(i) theset S = {vi} n 2 +1 i=1 isminimal-global-offensivealliancefor G; (ii) Γ= n 2 +1 for G; (iii) Γs =min{Σs∈S Σ3 i=1σi(s)}S={vi } n 2 +1 i=1 for G;

(iv) thesets S = {vi} n 2 +1 i=1 areonlyminimal-global-offensivealliancesfor G.

Proposition3.60. Let G bea m-familyofevencompletegraphswithcommonneutrosophic vertexset.Then

(i) theset S = {vi} n 2 i=1 isminimal-global-offensivealliancefor G;

(ii) Γ= n 2 for G;

(iii) Γs =min{Σs∈S Σ3 i=1σi(s)}S={vi } n 2 i=1 for G;

(iv) thesets S = {vi} n 2 i=1 areonlyminimal-global-offensivealliancesfor G.

3.6. GlobalPowerfulAlliance

Definition3.61. Let SHG =(G ⊆ P (V ),E ⊆ P (V ))beaneutrosophicSuperHyperGraph. Then HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

(i) aset S ofverticesiscalled t-offensivealliance if ∀a ∈ V \ S, |Ns(a) ∩ S|−|Ns(a) ∩ (V \ S)| >t;

(ii) at-offensiveallianceiscalled global-offensivealliance if t =0; (iii) aset S ofverticesiscalled t-defensivealliance if ∀a ∈ S, |Ns(a) ∩ S|−|Ns(a) ∩ (V \ S)| <t;

(iv) at-defensiveallianceiscalled global-defensivealliance if t =0; (v) aset S ofverticesiscalled t-powerfulalliance ifit’sbotht-offensiveallianceand (t-2)-defensivealliance;

(vi) at-powerfulallianceiscalled global-powerfulalliance if t =0; (vii) ∀S ⊆ S,S isglobal-powerfulalliancebut S isn’tglobal-powerfulalliance.Then S is called minimal-global-powerfulalliance; (viii) minimal-global-powerful-alliancenumber of SHG is S isaminimal-global-powerfulalliance. |S| andit’sdenotedbyΓ; (ix) minimal-global-powerful-alliance-neutrosophicnumber of SHG is S isaminimal-global-offensivealliance. Σs∈S Σ3 i=1σi(s) andit’sdenotedbyΓs

Proposition3.62. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beastrongneutrosophicSuperHyperGraph.Thenfollowingstatementshold;

(i) if s ≥ t andaset S ofverticesist-defensivealliance,then S iss-defensivealliance; (ii) if s ≤ t andaset S ofverticesist-offensivealliance,then S iss-offensivealliance.

Proposition3.63. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) beastrongneutrosophicSuperHyperGraph.Thenfollowingstatementshold;

(i) if s ≥ t +2 andaset S ofverticesist-defensivealliance,then S iss-powerfulalliance; (ii) if s ≤ t andaset S ofverticesist-offensivealliance,then S ist-powerfulalliance.

Proposition3.64. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) bear-regular-strong-neutrosophic SuperHyperGraph.Thenfollowingstatementshold;

(i) if ∀a ∈ S, |Ns(a) ∩ S| < r 2 +1, then SHG =(G ⊆ P (V ),E ⊆ P (V )) is2-defensive alliance;

(ii) if ∀a ∈ V \S, |Ns(a)∩S| > r 2 +1, then SHG =(G ⊆ P (V ),E ⊆ P (V )) is2-offensive alliance;

HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

(iii) if ∀a ∈ S, |Ns(a) ∩ V \ S| =0, then SHG =(G ⊆ P (V ),E ⊆ P (V )) isr-defensive alliance;

(iv) if ∀a ∈ V \ S, |Ns(a) ∩ V \ S| =0, then SHG =(G ⊆ P (V ),E ⊆ P (V )) isr-offensive alliance.

Proposition3.65. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) bear-regular-strong-neutrosophic SuperHyperGraph.Thenfollowingstatementshold;

(i) ∀a ∈ S, |Ns(a) ∩ S| < r 2 +1 if SHG =(G ⊆ P (V ),E ⊆ P (V )) is2-defensive alliance;

(ii) ∀a ∈ V \ S, |Ns(a) ∩ S| > r 2 +1 if SHG =(G ⊆ P (V ),E ⊆ P (V )) is2-offensive alliance;

(iii) ∀a ∈ S, |Ns(a) ∩ V \ S| =0 if SHG =(G ⊆ P (V ),E ⊆ P (V )) isr-defensivealliance; (iv) ∀a ∈ V \ S, |Ns(a) ∩ V \ S| =0 if SHG =(G ⊆ P (V ),E ⊆ P (V )) isr-offensive alliance.

Proposition3.66. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) bear-regular-strong-neutrosophic SuperHyperGraphwhichiscomplete.Thenfollowingstatementshold;

(i) ∀a ∈ S, |Ns(a) ∩ S| < O−1 2 +1 if SHG =(G ⊆ P (V ),E ⊆ P (V )) is2-defensive alliance;

(ii) ∀a ∈ V \ S, |Ns(a) ∩ S| > O−1 2 +1 if SHG =(G ⊆ P (V ),E ⊆ P (V )) is2-offensive alliance; (iii) ∀a ∈ S, |Ns(a) ∩ V \ S| =0 if SHG =(G ⊆ P (V ),E ⊆ P (V )) is (O− 1)-defensive alliance;

(iv) ∀a ∈ V \ S, |Ns(a) ∩ V \ S| =0 if SHG =(G ⊆ P (V ),E ⊆ P (V )) is (O− 1)-offensive alliance.

Proposition3.67. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) bear-regular-strong-neutrosophic SuperHyperGraphwhichiscomplete.Thenfollowingstatementshold;

(i) if ∀a ∈ S, |Ns(a) ∩S| < O−1 2 +1, then SHG =(G ⊆ P (V ),E ⊆ P (V )) is2-defensive alliance;

(ii) if ∀a ∈ V \ S, |Ns(a) ∩ S| > O−1 2 +1, then SHG =(G ⊆ P (V ),E ⊆ P (V )) is 2-offensivealliance;

(iii) if ∀a ∈ S, |Ns(a)∩V \S| =0, then SHG =(G ⊆ P (V ),E ⊆ P (V )) is (O−1)-defensive alliance;

(iv) if ∀a ∈ V \ S, |Ns(a) ∩ V \ S| =0, then SHG =(G ⊆ P (V ),E ⊆ P (V )) is (O− 1)offensivealliance.

Proposition3.68. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) bear-regular-strong-neutrosophic SuperHyperGraphwhichiscycle.Thenfollowingstatementshold;

HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph

(i) ∀a ∈ S, |Ns(a) ∩ S| < 2 if SHG =(G ⊆ P (V ),E ⊆ P (V )) is2-defensivealliance; (ii) ∀a ∈ V \ S, |Ns(a) ∩ S| > 2 if SHG =(G ⊆ P (V ),E ⊆ P (V )) is2-offensivealliance; (iii) ∀a ∈ S, |Ns(a) ∩ V \ S| =0 if SHG =(G ⊆ P (V ),E ⊆ P (V )) is2-defensivealliance; (iv) ∀a ∈ V \ S, |Ns(a) ∩ V \ S| =0 if SHG =(G ⊆ P (V ),E ⊆ P (V )) is2-offensive alliance.

Proposition3.69. Let SHG =(G ⊆ P (V ),E ⊆ P (V )) bear-regular-strong-neutrosophic SuperHyperGraphwhichiscycle.Thenfollowingstatementshold; (i) if ∀a ∈ S, |Ns(a)∩S| < 2, then SHG =(G ⊆ P (V ),E ⊆ P (V )) is2-defensivealliance; (ii) if ∀a ∈ V \ S, |Ns(a) ∩ S| > 2, then SHG =(G ⊆ P (V ),E ⊆ P (V )) is2-offensive alliance;

(iii) if ∀a ∈ S, |Ns(a) ∩ V \ S| =0, then SHG =(G ⊆ P (V ),E ⊆ P (V )) is2-defensive alliance; (iv) if ∀a ∈ V \ S, |Ns(a) ∩ V \ S| =0, then SHG =(G ⊆ P (V ),E ⊆ P (V )) is2-offensive alliance.

References

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[7] Shannon,A.;Atanassov,K.T.Afirststeptoatheoryoftheintuitionisticfuzzygraphs,Proceedingof FUBEST(Lakov,D.,Ed.)Sofia1994,59-61.

[8] Smarandache,F.AUnifyingfieldinlogicsneutrosophy:Neutrosophicprobability,setandlogic,Rehoboth: AmericanResearchPress1998.

[9] Smarandache,F.ExtensionofHyperGraphton-SuperHyperGraphandtoPlithogenicnSuperHyperGraph,andExtensionofHyperAlgebraton-ary(Classical-/Neutro-/Anti-)HyperAlgebraNeutrosophicSetsandSystems,2020,33290-296(http://fs.unm.edu/NSS/n-SuperHyperGraph-nHyperAlgebra.pdf).

[10] Wang,H.;Smarandache,F.;Zhang,Y.;Sunderraman,R.Single-valuedneutrosophicsets,Multispaceand Multistructure2010,4,410-413.

[11] Zadeh,L.A.Fuzzysets,InformationandControl1965,8,338-353.

Received: Dec. 5, 2021. Accepted: April 3, 2022.

HenryGarrett,PropertiesofSuperHyperGraphandNeutrosophicSuperHyperGraph