0037 | Matching Number in Neutrosophic Graphs

In this book, some notions are introduced about “Matching Number in Neutro- sophic Graphs.” Three chapters are devised as “Common Notions”, “Modified Notions” and “Extended Notions”. Three manuscripts are cited as the references of these chapters which are my 58th, 60th, and 61st manuscripts. I’ve used my 58th, 60th, and 61st manuscripts to write this book.

In first chapter, there are some points as follow. New setting is introduced to study matching number and matching neutrosophic-number arising from edges. Being endpoints for two edges, simultaneously is key type of approach to have these notions namely neutrosophic matching number and matching neutrosophic-number. Two numbers are obtained but now both settings leads to approach is on demand which is finding biggest set which have any edges which have no common edge inside set. Let NTG : (V,E,σ,μ) be a neutrosophic graph. Then matching number M(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is maximum cardinality of a set S of edges such that every two edges of S don’t have any vertex in common; matching neutrosophic-number Mn(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is maximum neutrosophic car- dinality of a set S of edges such that every two edges of S don’t have any vertex in common. As concluding results, there are some statements, remarks, examples and clarifications about some classes of neutrosophic graphs namely path-neutrosophic graphs, cycle-neutrosophic graphs, complete-neutrosophic graphs, star-neutrosophic graphs, complete-bipartite-neutrosophic graphs and complete-t-partite-neutrosophic graphs. The clarifications are also presented in both sections “Setting of Neutrosophic Matching Number,” and “Setting of Matching Neutrosophic-Number,” for introduced results and used classes. Neutrosophic number is reused in this way. It’s applied to use the type of neutrosophic number in the way that, three values of an edge are used and they’ve same share to construct this number to compare with other edges. Summation of three values of edge makes one number and applying it to a comparison. This approach facilitates identifying edges which form matching number and matching neutrosophic-number arising from being specific edge. In both settings, some classes of well-known neutrosophic graphs are studied. Some clarifications for each result and each definition are provided. Using basic set to extend this set to set of all edges has key role to have these notions in the form of matching number and matching neutrosophic-number arising from edges. The cardinality of a set has eligibility to neutrosophic matching number but the neutrosophic cardinality of a set has eligibility to call matching neutrosophic-number. Some results get more frameworks and perspective about

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Abstract

these definitions. The way in that, two edges have no connection with each other, opens the way to do some approaches. An edge could affect on other edge but there’s no usage of vertices. These notions are applied into neutrosophic graphs as individuals but not family of them as drawbacks for these notions. Finding special neutrosophic graphs which are well-known, is an open way to pursue this study. Some problems are proposed to pursue this study. Basic familiarities with graph theory and neutrosophic graph theory are proposed for this chapter.

In second chapter, there are some points as follow. New setting is introduced to study matching polynomial and matching polynomial neutrosophic-number arising from edges. Being endpoints for two edges, simultaneously is key type of approach to have these notions namely neutrosophic matching polynomial and matching polynomial neutrosophic-number. Two polynomials are obtained but now both settings leads to approach is on demand which is finding coef- ficients and powers. Let NTG : (V,E,σ,μ) be a neutrosophic graph. Then matching polynomial M(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is a polynomial where the coefficients of the terms of the matching polynomial represent the number of sets of independent edges of various cardinalities in G; matching polynomial neutrosophic-number Mn(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is a polynomial where the coefficients of the terms of the matching polynomial represent the number of sets of independent edges of various neutrosophic cardinalities in G. As concluding results, there are some statements, remarks, examples and clarifications about some classes of neut- rosophic graphs namely path-neutrosophic graphs, cycle-neutrosophic graphs, complete-neutrosophic graphs, star-neutrosophic graphs, complete-bipartite- neutrosophic graphs and complete-t-partite-neutrosophic graphs. The clarific- ations are also presented in both sections “Setting of Neutrosophic Matching Polynomial,” and “Setting of Matching Polynomial Neutrosophic-Number,” for introduced results and used classes. Neutrosophic number is reused in this way. It’s applied to use the type of neutrosophic number in the way that, three values of an edge are used and they’ve same share to construct this number to compare with other edges. Summation of three values of edge makes one number and applying it to a comparison. This approach facilitates identifying edges which form matching polynomial and matching polynomial neutrosophic- number arising from being specific edge. In both settings, some classes of well-known neutrosophic graphs are studied. Some clarifications for each result and each definition are provided. Using basic set to extend this set to set of all edges has key role to have these notions in the form of matching polynomial and matching polynomial neutrosophic-number arising from edges. The cardinality of a set and the number of these sets have eligibility to neutrosophic matching polynomial but the neutrosophic cardinality of a set the number of these sets have eligibility to call matching polynomial neutrosophic-number. Some results get more frameworks and perspective about these definitions. The way in that, two edges have no connection with each other, opens the way to do some approaches. An edge could affect on other edge but there’s no usage of vertices. These notions are applied into neutrosophic graphs as individuals but not family of them as drawbacks for these notions. Finding special neutrosophic graphs which are well-known, is an open way to pursue this study. Some problems are proposed to pursue this study. Basic familiarities with graph theory and neutrosophic graph theory are proposed for this chapter.

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In third chapter, there are some points as follow. New setting is introduced to study e-matching number, e-matching neutrosophic-number, e-matching polynomial and e-matching polynomial neutrosophic-number arising from edges and its corresponded vertices. Being endpoints for two edges, simultaneously is key type of approach to have these notions namely e-matching number, e-matching neutrosophic-number, e-matching polynomial and e-matching poly- nomial neutrosophic-number arising from edges and its corresponded vertices. Two numbers and two polynomials are obtained but now both settings leads to approach is on demand which is finding biggest set and counting k-set in the terms of vertices and edges, which have edges which have no common endpoint inside either set or corresponded set. Let NTG : (V,E,σ,μ) be a neutrosophic graph. Then e-matching number M(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is maximum cardinality of a set S containing en- dpoints of edges such that every two edges of S don’t have any vertex in common; e-matching neutrosophic-number Mn(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is maximum neutrosophic cardinality of a set S containing endpoints of edges such that every two edges of S don’t have any vertex in common. Let NTG : (V,E,σ,μ) be a neutrosophic graph. Then e-matching polynomial M(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is a polyno- mial where the coefficients of the terms of the e-matching polynomial represent the number of sets of endpoints of independent edges of various cardinalities in G. e-matching polynomial neutrosophic-number Mn(NTG) for a neutrosophic graph NTG : (V,E,σ,μ) is a polynomial where the coefficients of the terms of the e-matching polynomial represent the number of sets of endpoints of independent edges of various neutrosophic cardinalities in G. As concluding results, there are some statements, remarks, examples and clarifications about some classes of neutrosophic graphs namely path-neutrosophic graphs, cycle- neutrosophic graphs, complete-neutrosophic graphs, star-neutrosophic graphs, complete-bipartite-neutrosophic graphs and complete-t-partite-neutrosophic graphs. The clarifications are also presented in both sections “Setting of Neutrosophic e-Matching (Polynomial) Number,” and “Setting of e-Matching (Polynomial) Neutrosophic-Number,” for introduced results and used classes. Neutrosophic number is reused in this way. It’s applied to use the type of neut- rosophic number in the way that, three values of an edge are used and they’ve same share to construct this number to compare with other edges. Summation of three values of edge makes one number and applying it to a comparison. This approach facilitates identifying edges which form e-matching number, e-matching neutrosophic-number, e-matching polynomial and e-matching poly- nomial neutrosophic-number arising from edges and its corresponded vertices. In both settings, some classes of well-known neutrosophic graphs are studied. Some clarifications for each result and each definition are provided. Using basic set to extend this set to set of all edges has key role to have these notions in the form of e-matching number, e-matching neutrosophic-number, e-matching polynomial and e-matching polynomial neutrosophic-number arising from edges and its corresponded vertices. The cardinality of a set has eligibility to neutro- sophic e-matching (polynomial) number but the neutrosophic cardinality of a set has eligibility to call e-matching (polynomial) neutrosophic-number. Some results get more frameworks and perspective about these definitions. The way in that, two edges have no connection with each other, opens the way to do some approaches. An edge could affect on other edge but there’s no usage of vertices.

iii

Abstract

These notions are applied into neutrosophic graphs as individuals but not family of them as drawbacks for these notions. Finding special neutrosophic graphs which are well-known, is an open way to pursue this study. Some problems are proposed to pursue this study. Basic familiarities with graph theory and neutrosophic graph theory are proposed for this chapter.

The following references are cited by chapters.
[Ref1] Henry Garrett, “Matching Number in Neutrosophic Graphs”, ResearchG- ate 2022 (doi: 10.13140/RG.2.2.18609.86882).
[Ref2] Henry Garrett, “Matching Polynomials in Neutrosophic Graphs”, Re- searchGate 2022 (doi: 10.13140/RG.2.2.33630.72002).
[Ref3] Henry Garrett, “e-Matching Number and e-Matching Polynomials in Neutrosophic Graphs”, ResearchGate 2022 (doi: 10.13140/RG.2.2.32516.60805).

Three chapters are devised as “Common Notions”, “Modified Notions” and “Extended Notions”.