Zero Forcing Number in Neutrosophic Graphs

Zero Forcing Number in Neutrosophic Graphs
Author: Dr. Henry Garrett
Publisher: Dr. Henry Garrett
Total Pages: 252
Release: 2023-02-01
Genre: Mathematics
ISBN:
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In this research book, there are some research chapters on “Zero Forcing Number in Neutrosophic Graphs”. With researches on the basic properties, the research book starts to make Zero Forcing Number in Neutrosophic Graphs more understandable. Some studies and researches about neutrosophic graphs, are proposed as book in the following by Henry Garrett (2022) which is indexed by Google Scholar and has more than 2498 readers in Scribd. It’s titled “Beyond Neutrosophic Graphs” and published by Ohio: E-publishing: Educational Publisher 1091 West 1st Ave Grandview Heights, Ohio 43212 United State. This research book covers different types of notions and settings in neutrosophic graph theory and neutrosophic SuperHyperGraph theory. [Ref] Henry Garrett, (2022). “Beyond Neutrosophic Graphs”, Ohio: E-publishing: Educational Publisher 1091 West 1st Ave Grandview Heights, Ohio 43212 United States. ISBN: 978-1-59973-725-6 (http://fs.unm.edu/BeyondNeutrosophicGraphs.pdf). Also, some studies and researches about neutrosophic graphs, are proposed as book in the following by Henry Garrett (2022) which is indexed by Google Scholar and has more than 3218 readers in Scribd. It’s titled “Neutrosophic Duality” and published by Florida: GLOBAL KNOWLEDGE - Publishing House 848 Brickell Ave Ste 950 Miami, Florida 33131 United States. This research book presents different types of notions SuperHyperResolving and SuperHyperDominating in the setting of duality in neutrosophic graph theory and neutrosophic SuperHyperGraph theory. This research book has scrutiny on the complement of the intended set and the intended set, simultaneously. It’s smart to consider a set but acting on its complement that what’s done in this research book which is popular in the terms of high readers in Scribd. \section{Background} There are some researches covering the topic of this research. In what follows, there are some discussion and literature reviews about them. \\ First article is titled ``properties of SuperHyperGraph and neutrosophic SuperHyperGraph'' in \textbf{Ref.} \cite{HG1} by Henry Garrett (2022). It's first step toward the research on neutrosophic SuperHyperGraphs. This research article is published on the journal ``Neutrosophic Sets and Systems'' in issue 49 and the pages 531-561. In this research article, different types of notions like dominating, resolving, coloring, Eulerian(Hamiltonian) neutrosophic path, n-Eulerian(Hamiltonian) neutrosophic path, zero forcing number, zero forcing neutrosophic- number, independent number, independent neutrosophic-number, clique number, clique neutrosophic-number, matching number, matching neutrosophic-number, girth, neutrosophic girth, 1-zero-forcing number, 1-zero- forcing neutrosophic-number, failed 1-zero-forcing number, failed 1-zero-forcing neutrosophic-number, global- offensive alliance, t-offensive alliance, t-defensive alliance, t-powerful alliance, and global-powerful alliance are defined in SuperHyperGraph and neutrosophic SuperHyperGraph. Some Classes of SuperHyperGraph and Neutrosophic SuperHyperGraph are cases of research. Some results are applied in family of SuperHyperGraph and neutrosophic SuperHyperGraph. Thus this research article has concentrated on the vast notions and introducing the majority of notions. \\ The seminal paper and groundbreaking article is titled ``neutrosophic co-degree and neutrosophic degree alongside chromatic numbers in the setting of some classes related to neutrosophic hypergraphs'' in \textbf{Ref.} \cite{HG2} by Henry Garrett (2022). In this research article, a novel approach is implemented on SuperHyperGraph and neutrosophic SuperHyperGraph based on general forms without using neutrosophic classes of neutrosophic SuperHyperGraph. It's published in prestigious and fancy journal is entitled “Journal of Current Trends in Computer Science Research (JCTCSR)” with abbreviation ``J Curr Trends Comp Sci Res'' in volume 1 and issue 1 with pages 06-14. The research article studies deeply with choosing neutrosophic hypergraphs instead of neutrosophic SuperHyperGraph. It's the breakthrough toward independent results based on initial background. \\ The seminal paper and groundbreaking article is titled ``Super Hyper Dominating and Super Hyper Resolving on Neutrosophic Super Hyper Graphs and Their Directions in Game Theory and Neutrosophic Super Hyper Classes'' in \textbf{Ref.} \cite{HG3} by Henry Garrett (2022). In this research article, a novel approach is implemented on SuperHyperGraph and neutrosophic SuperHyperGraph based on fundamental SuperHyperNumber and using neutrosophic SuperHyperClasses of neutrosophic SuperHyperGraph. It's published in prestigious and fancy journal is entitled “Journal of Mathematical Techniques and Computational Mathematics(JMTCM)” with abbreviation ``J Math Techniques Comput Math'' in volume 1 and issue 3 with pages 242-263. The research article studies deeply with choosing directly neutrosophic SuperHyperGraph and SuperHyperGraph. It's the breakthrough toward independent results based on initial background and fundamental SuperHyperNumbers. \\ In some articles are titled ``0039 | Closing Numbers and Super-Closing Numbers as (Dual)Resolving and (Dual)Coloring alongside (Dual)Dominating in (Neutrosophic)n-SuperHyperGraph'' in \textbf{Ref.} \cite{HG4} by Henry Garrett (2022), ``0049 | (Failed)1-Zero-Forcing Number in Neutrosophic Graphs'' in \textbf{Ref.} \cite{HG5} by Henry Garrett (2022), ``Extreme SuperHyperClique as the Firm Scheme of Confrontation under Cancer’s Recognition as the Model in The Setting of (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG6} by Henry Garrett (2022), ``Uncertainty On The Act And Effect Of Cancer Alongside The Foggy Positions Of Cells Toward Neutrosophic Failed SuperHyperClique inside Neutrosophic SuperHyperGraphs Titled Cancer’s Recognition'' in \textbf{Ref.} \cite{HG7} by Henry Garrett (2022), ``Neutrosophic Version Of Separates Groups Of Cells In Cancer’s Recognition On Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG8} by Henry Garrett (2022), ``The Shift Paradigm To Classify Separately The Cells and Affected Cells Toward The Totality Under Cancer’s Recognition By New Multiple Definitions On the Sets Polynomials Alongside Numbers In The (Neutrosophic) SuperHyperMatching Theory Based on SuperHyperGraph and Neutrosophic SuperHyperGraph'' in \textbf{Ref.} \cite{HG9} by Henry Garrett (2022), ``Breaking the Continuity and Uniformity of Cancer In The Worst Case of Full Connections With Extreme Failed SuperHyperClique In Cancer’s Recognition Applied in (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG10} by Henry Garrett (2022), ``Neutrosophic Failed SuperHyperStable as the Survivors on the Cancer’s Neutrosophic Recognition Based on Uncertainty to All Modes in Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG11} by Henry Garrett (2022), ``Extremism of the Attacked Body Under the Cancer's Circumstances Where Cancer's Recognition Titled (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG12} by Henry Garrett (2022), ``(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG13} by Henry Garrett (2022), ``Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints'' in \textbf{Ref.} \cite{HG14} by Henry Garrett (2022), ``Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction To Use Neutrosophic SuperHyperGraphs on Cancer’s Neutrosophic Recognition And Beyond'' in \textbf{Ref.} \cite{HG15} by Henry Garrett (2022), ``(Neutrosophic) SuperHyperStable on Cancer’s Recognition by Well- SuperHyperModelled (Neutrosophic) SuperHyperGraphs '' in \textbf{Ref.} \cite{HG16} by Henry Garrett (2022), ``Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints'' in \textbf{Ref.} \cite{HG12} by Henry Garrett (2022), ``Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG17} by Henry Garrett (2022), ``Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints'' in \textbf{Ref.} \cite{HG18} by Henry Garrett (2022),``(Neutrosophic) SuperHyperModeling of Cancer’s Recognitions Featuring (Neutrosophic) SuperHyperDefensive SuperHyperAlliances'' in \textbf{Ref.} \cite{HG19} by Henry Garrett (2022), ``(Neutrosophic) SuperHyperAlliances With SuperHyperDefensive and SuperHyperOffensive Type-SuperHyperSet On (Neutrosophic) SuperHyperGraph With (Neutrosophic) SuperHyperModeling of Cancer’s Recognitions And Related (Neutrosophic) SuperHyperClasses'' in \textbf{Ref.} \cite{HG20} by Henry Garrett (2022), ``SuperHyperGirth on SuperHyperGraph and Neutrosophic SuperHyperGraph With SuperHyperModeling of Cancer’s Recognitions'' in \textbf{Ref.} \cite{HG21} by Henry Garrett (2022), ``Some SuperHyperDegrees and Co-SuperHyperDegrees on Neutrosophic SuperHyperGraphs and SuperHyperGraphs Alongside Applications in Cancer’s Treatments'' in \textbf{Ref.} \cite{HG22} by Henry Garrett (2022), ``SuperHyperDominating and SuperHyperResolving on Neutrosophic SuperHyperGraphs And Their Directions in Game Theory and Neutrosophic SuperHyperClasses'' in \textbf{Ref.} \cite{HG23} by Henry Garrett (2022), ``SuperHyperMatching By (R-)Definitions And Polynomials To Monitor Cancer’s Recognition In Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG24} by Henry Garrett (2023), ``The Focus on The Partitions Obtained By Parallel Moves In The Cancer's Extreme Recognition With Different Types of Extreme SuperHyperMatching Set and Polynomial on (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG25} by Henry Garrett (2023), ``Extreme Failed SuperHyperClique Decides the Failures on the Cancer's Recognition in the Perfect Connections of Cancer's Attacks By SuperHyperModels Named (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG26} by Henry Garrett (2023), ``Indeterminacy On The All Possible Connections of Cells In Front of Cancer's Attacks In The Terms of Neutrosophic Failed SuperHyperClique on Cancer's Recognition called Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG27} by Henry Garrett (2023), ``Perfect Directions Toward Idealism in Cancer's Neutrosophic Recognition Forwarding Neutrosophic SuperHyperClique on Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG28} by Henry Garrett (2023), ``Demonstrating Complete Connections in Every Embedded Regions and Sub-Regions in the Terms of Cancer's Recognition and (Neutrosophic) SuperHyperGraphs With (Neutrosophic) SuperHyperClique'' in \textbf{Ref.} \cite{HG29} by Henry Garrett (2023), ``Different Neutrosophic Types of Neutrosophic Regions titled neutrosophic Failed SuperHyperStable in Cancer’s Neutrosophic Recognition modeled in the Form of Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG30} by Henry Garrett (2023), ``Using the Tool As (Neutrosophic) Failed SuperHyperStable To SuperHyperModel Cancer's Recognition Titled (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG31} by Henry Garrett (2023), ``Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints'' in \textbf{Ref.} \cite{HG32} by Henry Garrett (2023), ``(Neutrosophic) SuperHyperStable on Cancer’s Recognition by Well-SuperHyperModelled (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG33} by Henry Garrett (2023), ``Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction To Use Neutrosophic SuperHyperGraphs on Cancer’s Neutrosophic Recognition And Beyond'' in \textbf{Ref.} \cite{HG34} by Henry Garrett (2022), ``(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG35} by Henry Garrett (2022), ``Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG36} by Henry Garrett (2022), ``Basic Neutrosophic Notions Concerning SuperHyperDominating and Neutrosophic SuperHyperResolving in SuperHyperGraph'' in \textbf{Ref.} \cite{HG37} by Henry Garrett (2022), ``Initial Material of Neutrosophic Preliminaries to Study Some Neutrosophic Notions Based on Neutrosophic SuperHyperEdge (NSHE) in Neutrosophic SuperHyperGraph (NSHG)'' in \textbf{Ref.} \cite{HG38} by Henry Garrett (2022), there are some endeavors to formalize the basic SuperHyperNotions about neutrosophic SuperHyperGraph and SuperHyperGraph. \\ Some studies and researches about neutrosophic graphs, are proposed as book in \textbf{Ref.} \cite{HG39} by Henry Garrett (2022) which is indexed by Google Scholar and has more than 2732 readers in Scribd. It's titled ``Beyond Neutrosophic Graphs'' and published by Ohio: E-publishing: Educational Publisher 1091 West 1st Ave Grandview Heights, Ohio 43212 United State. This research book covers different types of notions and settings in neutrosophic graph theory and neutrosophic SuperHyperGraph theory. \\ Also, some studies and researches about neutrosophic graphs, are proposed as book in \textbf{Ref.} \cite{HG40} by Henry Garrett (2022) which is indexed by Google Scholar and has more than 3504 readers in Scribd. It's titled ``Neutrosophic Duality'' and published by Florida: GLOBAL KNOWLEDGE - Publishing House 848 Brickell Ave Ste 950 Miami, Florida 33131 United States. This research book presents different types of notions SuperHyperResolving and SuperHyperDominating in the setting of duality in neutrosophic graph theory and neutrosophic SuperHyperGraph theory. This research book has scrutiny on the complement of the intended set and the intended set, simultaneously. It's smart to consider a set but acting on its complement that what's done in this research book which is popular in the terms of high readers in Scribd. -- \begin{thebibliography}{595} \bibitem{HG1} Henry Garrett, ``\textit{Properties of SuperHyperGraph and Neutrosophic SuperHyperGraph}'', Neutrosophic Sets and Systems 49 (2022) 531-561 (doi: 10.5281/zenodo.6456413). (http://fs.unm.edu/NSS/NeutrosophicSuperHyperGraph34.pdf). (https://digitalrepository.unm.edu/nss\_journal/vol49/iss1/34). \bibitem{HG2} Henry Garrett, ``\textit{Neutrosophic Co-degree and Neutrosophic Degree alongside Chromatic Numbers in the Setting of Some Classes Related to Neutrosophic Hypergraphs}'', J Curr Trends Comp Sci Res 1(1) (2022) 06-14. \bibitem{HG3} Henry Garrett, ``\textit{Super Hyper Dominating and Super Hyper Resolving on Neutrosophic Super Hyper Graphs and Their Directions in Game Theory and Neutrosophic Super Hyper Classes}'', J Math Techniques Comput Math 1(3) (2022) 242-263. \bibitem{HG4} Garrett, Henry. ``\textit{0039 | Closing Numbers and Super-Closing Numbers as (Dual)Resolving and (Dual)Coloring alongside (Dual)Dominating in (Neutrosophic)n-SuperHyperGraph.}'' CERN European Organization for Nuclear Research - Zenodo, Nov. 2022. CERN European Organization for Nuclear Research, https://doi.org/10.5281/zenodo.6319942. https://oa.mg/work/10.5281/zenodo.6319942 \bibitem{HG5} Garrett, Henry. ``\textit{0049 | (Failed)1-Zero-Forcing Number in Neutrosophic Graphs.}'' CERN European Organization for Nuclear Research - Zenodo, Feb. 2022. CERN European Organization for Nuclear Research, https://doi.org/10.13140/rg.2.2.35241.26724. https://oa.mg/work/10.13140/rg.2.2.35241.26724 \bibitem{HG6} Henry Garrett, ``\textit{Extreme SuperHyperClique as the Firm Scheme of Confrontation under Cancer’s Recognition as the Model in The Setting of (Neutrosophic) SuperHyperGraphs}'', Preprints 2023, 2023010308 (doi: 10.20944/preprints202301.0308.v1). \bibitem{HG7} Henry Garrett, ``\textit{Uncertainty On The Act And Effect Of Cancer Alongside The Foggy Positions Of Cells Toward Neutrosophic Failed SuperHyperClique inside Neutrosophic SuperHyperGraphs Titled Cancer’s Recognition}'', Preprints 2023, 2023010282 (doi: 10.20944/preprints202301.0282.v1). \bibitem{HG8} Henry Garrett, ``\textit{Neutrosophic Version Of Separates Groups Of Cells In Cancer’s Recognition On Neutrosophic SuperHyperGraphs}'', Preprints 2023, 2023010267 (doi: 10.20944/preprints202301.0267.v1). \bibitem{HG9} Henry Garrett, ``\textit{The Shift Paradigm To Classify Separately The Cells and Affected Cells Toward The Totality Under Cancer’s Recognition By New Multiple Definitions On the Sets Polynomials Alongside Numbers In The (Neutrosophic) SuperHyperMatching Theory Based on SuperHyperGraph and Neutrosophic SuperHyperGraph}'', Preprints 2023, 2023010265 (doi: 10.20944/preprints202301.0265.v1). \bibitem{HG10} Henry Garrett, ``\textit{Breaking the Continuity and Uniformity of Cancer In The Worst Case of Full Connections With Extreme Failed SuperHyperClique In Cancer’s Recognition Applied in (Neutrosophic) SuperHyperGraphs}'', Preprints 2023, 2023010262,(doi: 10.20944/preprints202301.0262.v1). \bibitem{HG11} Henry Garrett, ``\textit{Neutrosophic Failed SuperHyperStable as the Survivors on the Cancer’s Neutrosophic Recognition Based on Uncertainty to All Modes in Neutrosophic SuperHyperGraphs}'', Preprints 2023, 2023010240 (doi: 10.20944/preprints202301.0240.v1). \bibitem{HG12} Henry Garrett, ``\textit{Extremism of the Attacked Body Under the Cancer's Circumstances Where Cancer's Recognition Titled (Neutrosophic) SuperHyperGraphs}'', Preprints 2023, 2023010224, (doi: 10.20944/preprints202301.0224.v1). \bibitem{HG13} Henry Garrett, ``\textit{(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs}'', Preprints 2023, 2023010105 (doi: 10.20944/preprints202301.0105.v1). \bibitem{HG14} Henry Garrett, ``\textit{Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints}'', Preprints 2023, 2023010088 (doi: 10.20944/preprints202301.0088.v1). \bibitem{HG15} Henry Garrett, ``\textit{Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction To Use Neutrosophic SuperHyperGraphs on Cancer’s Neutrosophic Recognition And Beyond}'', Preprints 2023, 2023010044 \bibitem{HG16} Henry Garrett, ``\textit{(Neutrosophic) SuperHyperStable on Cancer’s Recognition by Well- SuperHyperModelled (Neutrosophic) SuperHyperGraphs}'', Preprints 2023, 2023010043 (doi: 10.20944/preprints202301.0043.v1). \bibitem{HG17} Henry Garrett, \textit{``Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs''}, Preprints 2023, 2023010105 (doi: 10.20944/preprints202301.0105.v1). \bibitem{HG18} Henry Garrett, \textit{``Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints''}, Preprints 2023, 2023010088 (doi: 10.20944/preprints202301.0088.v1). \bibitem{HG19} Henry Garrett, \textit{``(Neutrosophic) SuperHyperModeling of Cancer’s Recognitions Featuring (Neutrosophic) SuperHyperDefensive SuperHyperAlliances''}, Preprints 2022, 2022120549 (doi: 10.20944/preprints202212.0549.v1). \bibitem{HG20} Henry Garrett, ``\textit{(Neutrosophic) SuperHyperAlliances With SuperHyperDefensive and SuperHyperOffensive Type-SuperHyperSet On (Neutrosophic) SuperHyperGraph With (Neutrosophic) SuperHyperModeling of Cancer’s Recognitions And Related (Neutrosophic) SuperHyperClasses}'', Preprints 2022, 2022120540 (doi: 10.20944/preprints202212.0540.v1). \bibitem{HG21} Henry Garrett, ``\textit{SuperHyperGirth on SuperHyperGraph and Neutrosophic SuperHyperGraph With SuperHyperModeling of Cancer’s Recognitions}'', Preprints 2022, 2022120500 (doi: 10.20944/preprints202212.0500.v1). \bibitem{HG22} Henry Garrett, ``\textit{Some SuperHyperDegrees and Co-SuperHyperDegrees on Neutrosophic SuperHyperGraphs and SuperHyperGraphs Alongside Applications in Cancer’s Treatments}'', Preprints 2022, 2022120324 (doi: 10.20944/preprints202212.0324.v1). \bibitem{HG23} Henry Garrett, ``\textit{SuperHyperDominating and SuperHyperResolving on Neutrosophic SuperHyperGraphs And Their Directions in Game Theory and Neutrosophic SuperHyperClasses}'', Preprints 2022, 2022110576 (doi: 10.20944/preprints202211.0576.v1). \bibitem{HG24} Henry Garrett,``\textit{SuperHyperMatching By (R-)Definitions And Polynomials To Monitor Cancer’s Recognition In Neutrosophic SuperHyperGraphs}'', ResearchGate 2023,(doi: 10.13140/RG.2.2.35061.65767). \bibitem{HG25} Henry Garrett,``\textit{The Focus on The Partitions Obtained By Parallel Moves In The Cancer's Extreme Recognition With Different Types of Extreme SuperHyperMatching Set and Polynomial on (Neutrosophic) SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.18494.15680). \bibitem{HG26} Henry Garrett,``\textit{Extreme Failed SuperHyperClique Decides the Failures on the Cancer's Recognition in the Perfect Connections of Cancer's Attacks By SuperHyperModels Named (Neutrosophic) SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.32530.73922). \bibitem{HG27} Henry Garrett,``\textit{Indeterminacy On The All Possible Connections of Cells In Front of Cancer's Attacks In The Terms of Neutrosophic Failed SuperHyperClique on Cancer's Recognition called Neutrosophic SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.15897.70243). \bibitem{HG28} Henry Garrett,``\textit{Perfect Directions Toward Idealism in Cancer's Neutrosophic Recognition Forwarding Neutrosophic SuperHyperClique on Neutrosophic SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.30092.80004). \bibitem{HG29} Henry Garrett,``\textit{Demonstrating Complete Connections in Every Embedded Regions and Sub-Regions in the Terms of Cancer's Recognition and (Neutrosophic) SuperHyperGraphs With (Neutrosophic) SuperHyperClique}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.23172.19849). \bibitem{HG30} Henry Garrett,``\textit{Different Neutrosophic Types of Neutrosophic Regions titled neutrosophic Failed SuperHyperStable in Cancer’s Neutrosophic Recognition modeled in the Form of Neutrosophic SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.17385.36968). \bibitem{HG31} Henry Garrett, ``\textit{Using the Tool As (Neutrosophic) Failed SuperHyperStable To SuperHyperModel Cancer's Recognition Titled (Neutrosophic) SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.28945.92007). \bibitem{HG32} Henry Garrett, ``\textit{Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.11447.80803). \bibitem{HG33} Henry Garrett, ``\textit{(Neutrosophic) SuperHyperStable on Cancer’s Recognition by Well-SuperHyperModelled (Neutrosophic) SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.35774.77123). \bibitem{HG34} Henry Garrett, ``\textit{Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction To Use Neutrosophic SuperHyperGraphs on Cancer’s Neutrosophic Recognition And Beyond}'', ResearchGate 2022, (doi: 10.13140/RG.2.2.36141.77287). \bibitem{HG35} Henry Garrett, ``\textit{(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs}'', ResearchGate 2022, (doi: 10.13140/RG.2.2.29430.88642). \bibitem{HG36} Henry Garrett, ``\textit{Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs}'', ResearchGate 2022, (doi: 10.13140/RG.2.2.11369.16487). \bibitem{HG37} Henry Garrett, \textit{``Basic Neutrosophic Notions Concerning SuperHyperDominating and Neutrosophic SuperHyperResolving in SuperHyperGraph''}, ResearchGate 2022 (doi: 10.13140/RG.2.2.29173.86244). \bibitem{HG38} Henry Garrett, ``\textit{Initial Material of Neutrosophic Preliminaries to Study Some Neutrosophic Notions Based on Neutrosophic SuperHyperEdge (NSHE) in Neutrosophic SuperHyperGraph (NSHG)}'', ResearchGate 2022 (doi: 10.13140/RG.2.2.25385.88160). \bibitem{HG39} Henry Garrett, (2022). ``\textit{Beyond Neutrosophic Graphs}'', Ohio: E-publishing: Educational Publisher 1091 West 1st Ave Grandview Heights, Ohio 43212 United States. ISBN: 979-1-59973-725-6 (http://fs.unm.edu/BeyondNeutrosophicGraphs.pdf). \bibitem{HG40} Henry Garrett, (2022). ``\textit{Neutrosophic Duality}'', Florida: GLOBAL KNOWLEDGE - Publishing House 848 Brickell Ave Ste 950 Miami, Florida 33131 United States. ISBN: 978-1-59973-743-0 (http://fs.unm.edu/NeutrosophicDuality.pdf). \end{thebibliography} [Ref] Henry Garrett, (2022). “Neutrosophic Duality”, Florida: GLOBAL KNOW- LEDGE - Publishing House 848 Brickell Ave Ste 950 Miami, Florida 33131 United States. ISBN: 978-1-59973-743-0 (http://fs.unm.edu/NeutrosophicDuality.pdf).

Zero Forcing Number

Zero Forcing Number
Author: Darren Daniel Row
Publisher:
Total Pages: 51
Release: 2011
Genre:
ISBN:
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Domination in Graphs

Domination in Graphs
Author: TeresaW. Haynes
Publisher: Routledge
Total Pages: 519
Release: 2017-11-22
Genre: Mathematics
ISBN: 1351454641
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""Presents the latest in graph domination by leading researchers from around the world-furnishing known results, open research problems, and proof techniques. Maintains standardized terminology and notation throughout for greater accessibility. Covers recent developments in domination in graphs and digraphs, dominating functions, combinatorial problems on chessboards, and more.

Inverse Problems and Zero Forcing for Graphs

Inverse Problems and Zero Forcing for Graphs
Author: Leslie Hogben
Publisher: American Mathematical Society
Total Pages: 302
Release: 2022-07-21
Genre: Mathematics
ISBN: 1470466554
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This book provides an introduction to the inverse eigenvalue problem for graphs (IEP-$G$) and the related area of zero forcing, propagation, and throttling. The IEP-$G$ grew from the intersection of linear algebra and combinatorics and has given rise to both a rich set of deep problems in that area as well as a breadth of “ancillary” problems in related areas. The IEP-$G$ asks a fundamental mathematical question expressed in terms of linear algebra and graph theory, but the significance of such questions goes beyond these two areas, as particular instances of the IEP-$G$ also appear as major research problems in other fields of mathematics, sciences and engineering. One approach to the IEP-$G$ is through rank minimization, a relevant problem in itself and with a large number of applications. During the past 10 years, important developments on the rank minimization problem, particularly in relation to zero forcing, have led to significant advances in the IEP-$G$. The monograph serves as an entry point and valuable resource that will stimulate future developments in this active and mathematically diverse research area.

Properties of the Zero Forcing Number

Properties of the Zero Forcing Number
Author: Kayla Denise Owens
Publisher:
Total Pages: 46
Release: 2009
Genre: Graph theory
ISBN:
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The zero forcing number is a graph parameter first introduced as a tool for solving the minimum rank problem, which is: Given a simple, undirected graph G, and a field F, let S(F, G) denote the set of all symmetric matrices A = [aij] with entires in F such that aij is not equal to 0 if and only if ij is an edge in G. Find the minimum possible rank of a matrix in S(F, G). It is known that the zero forcing number Z(G) provides an upper bound for the maximum nullity of a graph. I investigate properties of the zero forcing number, including its behavior under various graph operations.

Pearls in Graph Theory

Pearls in Graph Theory
Author: Nora Hartsfield
Publisher: Courier Corporation
Total Pages: 276
Release: 2013-04-15
Genre: Mathematics
ISBN: 0486315525
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Stimulating and accessible, this undergraduate-level text covers basic graph theory, colorings of graphs, circuits and cycles, labeling graphs, drawings of graphs, measurements of closeness to planarity, graphs on surfaces, and applications and algorithms. 1994 edition.

Zero Forcing Sets for Graphs

Zero Forcing Sets for Graphs
Author: Fatemeh Alinaghipour Taklimi
Publisher:
Total Pages:
Release: 2014
Genre:
ISBN:
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Cell-Free Massive MIMO

Cell-Free Massive MIMO
Author: Giovanni Interdonato
Publisher: Linköping University Electronic Press
Total Pages: 75
Release: 2020-09-09
Genre: Electronic books
ISBN: 9179298087
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The fifth generation of mobile communication systems (5G) is nowadays a reality. 5G networks are been deployed all over the world, and the first 5G-capable devices (e.g., smartphones, tablets, wearable, etc.) are already commercially available. 5G systems provide unprecedented levels of connectivity and quality of service (QoS) to cope with the incessant growth in the number of connected devices and the huge increase in data-rate demand. Massive MIMO (multiple-input multiple-output) technology plays a key role in 5G systems. The underlying principle of this technology is the use of a large number of co-located antennas at the base station, which coherently transmit/receive signals to/from multiple users. This signal co-processing at multiple antennas leads to manifold benefits: array gain, spatial diversity and spatial user multiplexing. These elements enable to meet the QoS requirements established for the 5G systems. The major bottleneck of massive MIMO systems as well as of any cellular network is the inter-cell interference, which affects significantly the cell-edge users, whose performance is already degraded by the path attenuation. To overcome these limitations and provide uniformly excellent service to all the users we need a more radical approach: we need to challenge the cellular paradigm. In this regard, cell-free massive MIMO constitutes the paradigm shift. In the cell-free paradigm, it is not the base station surrounded by the users, but rather it is each user being surrounded by smaller, simpler, serving base stations referred to as access points (APs). In such a system, each user experiences being in the cell-center, and it does not experience any cell boundaries. Hence, the terminology cell-free. As a result, users are not affected by inter-cell interference, and the path attenuation is significantly reduced due to the presence of many APs in their proximity. This leads to impressive performance. Although appealing from the performance viewpoint, the designing and implementation of such a distributed massive MIMO system is a challenging task, and it is the object of this thesis. More specifically, in this thesis we study: Paper A) The large potential of this promising technology in realistic indoor/outdoor scenarios while also addressing practical deployment issues, such as clock synchronization among APs, and cost-efficient implementations. We provide an extensive description of a cell-free massive MIMO system, emphasizing strengths and weaknesses, and pointing out differences and similarities with existing distributed multiple antenna systems, such as Coordinated MultiPoint (CoMP). Paper B) How to preserve the scalability of the system, by proposing a solution related to data processing, network topology and power control. We consider a realistic scenario where multiple central processing units serve disjoint subsets of APs, and compare the spectral efficiency provided by the proposed scalable framework with the canonical cell-free massive MIMO and CoMP. Paper C) How to improve the spectral efficiency (SE) in the downlink (DL), by devising two distributed precoding schemes, referred to as local partial zero-forcing (ZF) and local protective partial ZF, that provide an adaptable trade-off between interference cancelation and boosting of the desired signal, with no additional front-haul overhead, and that are implementable by APs with very few antennas. We derive closed-form expressions for the achievable SE under the assumption of independent Rayleigh fading channel, channel estimation error and pilot contamination. These closed-form expressions are then used to devise optimal max-min fairness power control. Paper D) How to further improve the SE by letting the user estimate the DL channel from DL pilots, instead of relying solely on the knowledge of the channel statistics. We derive an approximate closed-form expression of the DL SE for conjugate beamforming (CB), and assuming independent Rayleigh fading. This expression accounts for beamformed DL pilots, estimation errors and pilot contamination at both the AP and the user side. We devise a sequential convex approximation algorithm to globally solve the max-min fairness power control optimization problem, and a greedy algorithm for uplink (UL) and DL pilot assignment. The latter consists in jointly selecting the UL and DL pilot pair, for each user, that maximizes the smallest SE in the network. Paper E) A precoding scheme that is more suitable when only the channel statistics are available at the users, referred to as enhanced normalized CB. It consists in normalizing the precoding vector by its squared norm in order to reduce the fluctuations of the effective channel seen at the user, and thereby to boost the channel hardening. The performance achieved by this scheme is compared with the CB scheme with DL training (described in Paper D). Paper F) A maximum-likelihood-based method to estimate the channel statistics in the UL, along with an accompanying pilot transmission scheme, that is particularly useful in line-of-sight operation and in scenarios with resource constraints. Pilots are structurally phase-rotated over different coherence blocks to create an effective statistical distribution of the received pilot signal that can be efficiently exploited by the AP when performing the proposed estimation method. The overall conclusion is that cell-free massive MIMO is not a utopia, and a practical, distributed, scalable, high-performance system can be implemented. Today it represents a hot research topic, but tomorrow it might represent a key enabler for beyond-5G technology, as massive MIMO has been for 5G. La quinta generazione dei sistemi radiomobili cellulari (5G) è oggi una realtà. Le reti 5G si stanno diffondendo in tutto il mondo e i dispositivi 5G (ad esempio smartphones, tablets, indossabili, ecc.) sono già disponibili sul mercato. I sistemi 5G garantiscono livelli di connettività e di qualità di servizio senza precedenti, per fronteggiare l’incessante crescita del numero di dispositivi connessi alla rete e della domanda di dati ad alta velocità. La tecnologia Massive MIMO (multiple-input multiple-output) riveste un ruolo fondamentale nei sistemi 5G. Il principio alla base di questa tecnologia è l’impiego di un elevato numero di antenne collocate nella base station (stazione radio base) le quali trasmettono/ricevono segnali, in maniere coerente, a/da più terminali utente. Questo co-processamento del segnale da parte di più antenne apporta molteplici benefici: guadagno di array, diversità spaziale e multiplazione degli utenti nel dominio spaziale. Questi elementi consentono di raggiungere i requisiti di servizio stabiliti per i sistemi 5G. Tuttavia, il limite principale dei sistemi massive MIMO, così come di ogni rete cellulare, è rappresentato dalla interferenza inter-cella (ovvero l’interferenza tra aree di copertura gestite da diverse base stations), la quale riduce in modo significativo le performance degli utenti a bordo cella, già degradate dalle attenuazioni del segnale dovute alla considerevole distanza dalla base station. Per superare queste limitazioni e fornire una qualità del servizio uniformemente eccellente a tutti gli utenti, è necessario un approccio più radicale e guardare oltre il classico paradigma cellulare che caratterizza le attuali architetture di rete. A tal proposito, cell-free massive MIMO (massive MIMO senza celle) costituisce un cambio di paradigma: ogni utente è circondato e servito contemporaneamente da numerose, semplici e di dimensioni ridotte base stations, denominate access points (punti di accesso alla rete). Gli access points cooperano per servire tutti gli utenti nella loro area di copertura congiunta, eliminando l’interferenza inter-cella e il concetto stesso di cella. Non risentendo più dell’effetto “bordo-cella”, gli utenti possono usufruire di qualità di servizio e velocità dati eccellenti. Sebbene attraente dal punto di vista delle performance, l’implementazione di un tale sistema distribuito è una operazione impegnativa ed è oggetto di questa tesi. Piu specificatamente, questa tesi di dottorato tratta: Articolo A) L’enorme potenziale di questa promettente tecnologia in scenari realistici sia indoor che outdoor, proponendo anche delle soluzioni di implementazione flessibili ed a basso costo. Articolo B) Come preservare la scalabilità del sistema, proponendo soluzioni distribuite riguardanti il processamento e la condivisione dei dati, l’architettura di rete e l’allocazione di potenza, ovvero come ottimizzare i livelli di potenza trasmessa dagli access points per ridurre l’interferenza tra utenti e migliorare le performance. Articolo C) Come migliorare l’efficienza spettrale in downlink (da access point verso utente) proponendo due schemi di pre-codifica dei dati di trasmissione, denominati local partial zero-forcing (ZF) e local protective partial ZF, che forniscono un perfetto compromesso tra cancellazione dell’interferenza tra utenti ed amplificazione del segnale desiderato. Articolo D) Come migliorare l’efficienza spettrale in downlink permettendo al terminale utente di stimare le informazioni sulle condizioni istantanee del canale da sequenze pilota, piuttosto che basarsi su informazioni statistiche ed a lungo termine, come convenzionalmente previsto. Articolo E) In alternativa alla soluzione precedente, uno schema di pre-codifica che è più adatto al caso in cui gli utenti hanno a disposizione esclusivamente informazioni statistiche sul canale per poter effettuare la decodifica dei dati. Articolo F) Un metodo per permettere agli access points di stimare, in maniera rapida, le condizioni di canale su base statistica, favorito da uno schema di trasmissione delle sequenze pilota basato su rotazione di fase. Realizzare un sistema cell-free massive MIMO pratico, distribuito, scalabile e performante non è una utopia. Oggi questo concept rappresenta un argomento di ricerca interessante, attraente e stimolante ma in futuro potrebbe costituire un fattore chiave per le tecnologie post-5G, proprio come massive MIMO lo è stato per il 5G. Den femte generationens mobilkommunikationssystem (5G) är numera en verklighet. 5G-nätverk är utplacerade på ett flertal platser världen över och de första 5G-kapabla terminalerna (såsom smarta telefoner, surfplattor, kroppsburna apparater, etc.) är redan kommersiellt tillgängliga. 5G-systemen kan tillhandahålla tidigare oöverträffade nivåer av uppkoppling och servicekvalitet och är designade för en fortsatt oavbruten tillväxt i antalet uppkopplade apparater och ökande datataktskrav. Massiv MIMO-teknologi (eng: multiple-input multiple-output) spelar en nyckelroll i dagens 5G-system. Principen bakom denna teknik är användningen av ett stort antal samlokaliserade antenner vid basstationen, där alla antennerna sänder och tar emot signaler faskoherent till och från flera användare. Gemensam signalbehandling av många antennsignaler ger ett flertal fördelar, såsom hög riktverkan via lobformning, vilket leder till högre datatakter samt möjliggör att flera användare utnyttjar samma radioresurser via rumslig användarmultiplexering. Eftersom en signal kan gå genom flera olika, möjligen oberoende kanaler, så utsätts den för flera olika förändringar samtidigt. Denna mångfald ökar kvaliteten på signalen vid mottagaren och förbättrar radiolänkens robusthet och tillförlitlighet. Detta gör det möjligt att uppfylla de höga kraven på servicekvalitet som fastställts för 5G-systemen. Den största begränsningen för massiva MIMO-system såväl som för alla cellulära mobilnätverk, är störningar från andra celler som påverkar användare på cellkanten väsentligt, vars prestanda redan begränsas av sträckdämpningen på radiokanalen. För att övervinna dessa begränsningar och för att kunna tillhandahålla samma utmärkta servicekvalitet till alla användare behöver vi ett mer radikalt angreppssätt: vi måste utmana cellparadigmet. I detta avseende utgör cellfri massiv-MIMO teknik ett paradigmskifte. I cellfri massive-MIMO är utgångspunkten inte att basstationen är omgiven av användare som den betjänar, utan snarare att varje användare omges av basstationer som de betjänas av. Dessa basstationer, ofta mindre och enklare, kallas accesspunkter (AP). I ett sådant system upplever varje användare att den befinner sig i centrum av systemet och ingen användare upplever några cellgränser. Därav terminologin cellfri. Som ett resultat av detta påverkas inte användarna av inter-cellstörningar och sträckdämpningen reduceras kraftigt på grund av närvaron av många accesspunkter i varje användares närhet. Detta leder till imponerande prestanda. Även om det är tilltalande ur ett prestandaperspektiv så är utformningen och implementeringen av ett sådant distribuerat massivt MIMO-system en utmanande uppgift, och det är syftet med denna avhandling att studera detta. Mer specifikt studerar vi i denna avhandling: A) den mycket stora potentialen med denna teknik i realistiska inomhus- såväl som utomhusscenarier, samt hur man hanterar praktiska implementeringsproblem, såsom klocksynkronisering bland accesspunkter och kostnadseffektiva implementeringar; B) hur man ska uppnå skalbarhet i systemet genom att föreslå lösningar relaterade till databehandling, nätverkstopologi och effektkontroll; C) hur man ökar datahastigheten i nedlänken med hjälp av två nyutvecklade distribuerade överföringsmetoder som tillhandahåller en avvägning mellan störningsundertryckning och förstärkning av önskade signaler, utan att öka mängden intern signalering till de distribuerade accesspunkterna, och som kan implementeras i accesspunkter med mycket få antenner; D) hur man kan förbättra prestandan ytterligare genom att låta användaren estimera nedlänkskanalen med hjälp av nedlänkspiloter, istället för att bara förlita sig på kunskap om kanalstatistik; E) en överföringsmetod för nedlänk som är mer lämpligt när endast kanalstatistiken är tillgänglig för användarna. Prestandan som uppnås genom detta schema jämförs med en utökad variant av den nedlänk-pilotbaserade metoden (beskrivet i föregående punkt); F) en metod för att uppskatta kanalstatistiken i upplänken, samt en åtföljande pilotsändningsmetod, som är särskilt användbart vid direktvägsutbredning (line-of-sight) och i scenarier med resursbegränsningar. Den övergripande slutsatsen är att cellfri massiv MIMO inte är en utopi, och att ett distribuerat, skalbart, samt högpresterande system kan implementeras praktiskt. Idag representerar detta ett hett forskningsämne, men snart kan det visa sig vara en viktig möjliggörare för teknik bortom dagens system, på samma sätt som centraliserad massiv MIMO har varit för de nya 5G-systemen.

Hypergraph Theory

Hypergraph Theory
Author: Alain Bretto
Publisher: Springer Science & Business Media
Total Pages: 129
Release: 2013-04-17
Genre: Mathematics
ISBN: 3319000802
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This book provides an introduction to hypergraphs, its aim being to overcome the lack of recent manuscripts on this theory. In the literature hypergraphs have many other names such as set systems and families of sets. This work presents the theory of hypergraphs in its most original aspects, while also introducing and assessing the latest concepts on hypergraphs. The variety of topics, their originality and novelty are intended to help readers better understand the hypergraphs in all their diversity in order to perceive their value and power as mathematical tools. This book will be a great asset to upper-level undergraduate and graduate students in computer science and mathematics. It has been the subject of an annual Master's course for many years, making it also ideally suited to Master's students in computer science, mathematics, bioinformatics, engineering, chemistry, and many other fields. It will also benefit scientists, engineers and anyone else who wants to understand hypergraphs theory.

Fuzzy Graph Theory

Fuzzy Graph Theory
Author: Sunil Mathew
Publisher: Springer
Total Pages: 331
Release: 2017-12-30
Genre: Technology & Engineering
ISBN: 3319714074
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This book provides a timely overview of fuzzy graph theory, laying the foundation for future applications in a broad range of areas. It introduces readers to fundamental theories, such as Craine’s work on fuzzy interval graphs, fuzzy analogs of Marczewski’s theorem, and the Gilmore and Hoffman characterization. It also introduces them to the Fulkerson and Gross characterization and Menger’s theorem, the applications of which will be discussed in a forthcoming book by the same authors. This book also discusses in detail important concepts such as connectivity, distance and saturation in fuzzy graphs. Thanks to the good balance between the basics of fuzzy graph theory and new findings obtained by the authors, the book offers an excellent reference guide for advanced undergraduate and graduate students in mathematics, engineering and computer science, and an inspiring read for all researchers interested in new developments in fuzzy logic and applied mathematics.