Neutrosophic Connectivity

In this book, some notions are introduced about “Neutrosophic Connectivities.” Three chapters are devised as “Weakest Path Connectivities”, “Strongest Path Connectivities” and “Cycle Connectivities”. Three manuscripts are cited as the references of these chapters which are my 70th, 71st, and 72nd manuscripts. I’ve used my 70th, 71st, and 72nd manuscripts to write this book.

In first chapter, there are some points as follow. New setting is introduced to study weakest number and neutrosophic weakest number arising from length and strength of paths in neutrosophic graphs assigned to couple of vertices and to neutrosophic graphs. Forming neutrosophic paths from a sequence of consecutive vertices forming different types of paths in the terms of length and minimum values of edges forming it to get maximum number to assign to neutrosophic graphs and couple of vertices, is key type of approach to have these notions namely weakest number and neutrosophic weakest number arising from length and strength of paths in neutrosophic graphs assigned to couple of vertices and to neutrosophic graphs. Two numbers and two different paths are assigned to couple of vertices and to neutrosophic graphs, are obtained but now both settings lead to approach is on demand which is to compute and to find sequence of consecutive vertices forming different types of paths in the terms of length and minimum values of edges forming it to get max- imum number to assign to neutrosophic graphs and couple of vertices. Let NTG : (V,E,σ,μ) be a neutrosophic graph. Then a path from x to y is called weakest path if its length is maximum. This length is called weakest number amid x and y. The maximum number amid all vertices is called weakest number of NTG : (V,E,σ,μ) and it’s denoted by W(NTG); a path from x to y is called neutrosophic weakest path if its strength is μ(uv) which is less than all strengths of all paths from x to y where x,··· ,u,v,··· ,y is a path. This strength is called neutrosophic weakest number amid x and y. The maximum number amid all vertices is called neutrosophic weakest number of NTG : (V,E,σ,μ) and it’s denoted by Wn(NTG). As concluding results, there are some statements, remarks, examples and clarifications about some classes of strong neutrosophic graphs namely path-neutrosophic graphs, cycle-neutrosophic graphs, complete- neutrosophic graphs, star-neutrosophic graphs, complete-bipartite-neutrosophic graphs, complete-t-partite-neutrosophic graphs and wheel-neutrosophic graphs. The clarifications are also presented in both sections “Setting of Weakest Num- ber,” and “Setting of Neutrosophic Weakest Number,” for introduced results and used classes. This approach facilitates identifying vertices which form weakest number and neutrosophic weakest number arising from length and

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Abstract

strength of paths in neutrosophic graphs assigned to couple of vertices and to neutrosophic graphs. In both settings, some classes of well-known neutrosophic graphs are studied. Some clarifications for each result and each definition are provided. The value of an edge has eligibility to define weakest number and neutrosophic weakest number but the sequence has eligibility to define weakest path and neutrosophic weakest path. Some results get more frameworks and perspective about these definitions. The way in that, sequence of consecutive vertices forming different types of paths in the terms of length and minimum values of edges forming it to get maximum number to assign to neutrosophic graphs and couple of vertices, opens the way to do some approaches. 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. Neutrosophic path connectivity is applied in different settings and classes of neutrosophic graphs. 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 strongest number and neutrosophic strongest number arising from length and strength of paths in neutrosophic graphs assigned to couple of vertices and to neutrosophic graphs. Forming neutrosophic paths from a sequence of consecutive vertices forming different types of paths in the terms of length and maximum values of edges forming it to get minmum number to assign to neutrosophic graphs and couple of vertices, is key type of approach to have these notions namely strongest number and neutrosophic strongest number arising from length and strength of paths in neutrosophic graphs assigned to couple of vertices and to neutrosophic graphs. Two numbers and two differ- ent paths are assigned to couple of vertices and to neutrosophic graphs, are obtained but now both settings lead to approach is on demand which is to compute and to find sequence of consecutive vertices forming different types of paths in the terms of length and maximum values of edges forming it to get minimum number to assign to neutrosophic graphs and couple of vertices. Let NTG : (V,E,σ,μ) be a neutrosophic graph. Then a path from x to y is called strongest path if its length is minimum. This length is called strongest number amid x and y. The maximum number amid all vertices is called strongest number of NTG : (V,E,σ,μ) and it’s denoted by S(NTG); a path from x to y is called neutrosophic strongest path if its strength is μ(uv) which is greater than all strengths of all paths from x to y where x,··· ,u,v,··· ,y is a path. This strength is called neutrosophic strongest number amid x and y. The minimum number amid all vertices is called neutrosophic strongest number of NTG : (V,E,σ,μ) and it’s denoted by Sn(NTG). As concluding results, there are some statements, remarks, examples and clarifications about some classes of strong neutrosophic graphs namely strong-path-neutrosophic graphs, strong-cycle-neutrosophic graphs, strong-complete-neutrosophic graphs, strong-star-neutrosophic graphs, strong-complete-bipartite-neutrosophic graphs, strong-complete-t-partite-neutrosophic graphs and strong-wheel-neutrosophic graphs. The clarifications are also presented in both sections “Setting of strongest Number,” and “Setting of Neutrosophic strongest Number,” for intro- duced results and used classes. This approach facilitates identifying vertices which form strongest number and neutrosophic strongest number arising from length and strength of paths in neutrosophic graphs assigned to couple of ver-

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tices and to neutrosophic graphs. In both settings, some classes of well-known neutrosophic graphs are studied. Some clarifications for each result and each definition are provided. The value of an edge has eligibility to define strongest number and neutrosophic strongest number but the sequence has eligibility to define strongest path and neutrosophic strongest path. Some results get more frameworks and perspective about these definitions. The way in that, sequence of consecutive vertices forming different types of paths in the terms of length and maximum values of edges forming it to get minimum number to assign to neutrosophic graphs and couple of vertices, opens the way to do some approaches. 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. Neutrosophic path connectivity is applied in different settings and classes of neutrosophic graphs. Some problems are proposed to pursue this study. Basic familiarities with graph theory and neutrosophic graph theory are proposed for this chapter.

In third chapter, there are some points as follow. New setting is introduced to study connectivity number and neutrosophic connectivity number arising from length and strength of cycles in neutrosophic graphs assigned to one vertex and to neutrosophic graphs. Consider a vertex. Maximum strengths of all cycles based on that vertex is a number which is representative strength based on that vertex. Minimum number amid all representative numbers is called neutrosophic connectivity number. Forming cycles from a sequence of consecutive vertices to figure out different types of cycles in the terms of length and maximum values of edges to get minimum number to assign to neutrosophic graphs and a vertex, is key type of approach to have these notions namely connectivity number and neutrosophic connectivity number arising from length and strength of cycles in neutrosophic graphs assigned to one vertex and to neutrosophic graphs. Two numbers and two different cycles are assigned to a vertex and to a neutrosophic graph, are obtained but now both settings lead to approach is on demand which is to compute and to find sequence of consecutive vertices forming different types of cycles in the terms of length and maximum values of edges forming it to get minimum number to assign to a neutrosophic graph and to a vertex. Let NTG : (V,E,σ,μ) be a neutrosophic graph. Then a cycle based on x is called cyclic connectivity if its length is minimum. This length is called connectivity number based on x. The maximum number amid all vertices is called connectivity number of NTG : (V,E,σ,μ) and it’s denoted by C(NTG); a cycle based on x is called neutrosophic cyclic connectivity if its strength is is greater than all strengths of all cycles based on x. This strength is called neutrosophic connectivity number based on x. The minimum number amid all vertices is called neutrosophic connectivity number of NTG : (V,E,σ,μ) and it’s denoted by Cn(NTG). As concluding results, there are some statements, remarks, examples and clarifications about some classes of strong-neutrosophic graphs namely strong-path-neutrosophic graphs, strong-cycle-neutrosophic graphs, strong-complete-neutrosophic graphs, strong-star-neutrosophic graphs, strong-complete-bipartite-neutrosophic graphs, strong-complete-t-partite-neutrosophic graphs and strong-wheel-neutrosophic graphs. The clarifications are also presented in both sections “Setting of con- nectivity number,” and “Setting of neutrosophic connectivity number,” for introduced results and used classes. This approach facilitates identifying ver-

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Abstract

tices which form connectivity number and neutrosophic connectivity number arising from length and strength of cycles in neutrosophic graphs assigned to one vertex and to neutrosophic graphs. In both settings, some classes of well-known neutrosophic graphs are studied. Some clarifications for each result and each definition are provided. The value of an edge has eligibility to define connectivity number and neutrosophic connectivity number but the sequence has eligibility to define cyclic connectivity and neutrosophic cyclic connectivity. Some results get more frameworks and perspective about these definitions. The way in that, sequence of consecutive vertices forming different types of cycles in the terms of length and maximum values of edges forming it to get minimum number to assign to neutrosophic graphs and a vertex or in other words, the way in that, consider a vertex. Maximum strengths of all cycles based on that vertex is a number which is representative strength based on that vertex. Minimum number amid all representative numbers is called neutrosophic connectivity number, opens the way to do some approaches. 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. Neutrosophic cycle connectivity is applied to different settings and classes of neutrosophic graphs. 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, “Finding Longest Weakest Paths assigning num- bers to some Classes of Neutrosophic Graphs”, ResearchGate 2022 (doi: 10.13140/RG.2.2.27390.87361).
[Ref2] Henry Garrett, “Strongest Paths Defining Connectivities in Neutrosophic

Graphs”, ResearchGate 2022 (doi: 10.13140/RG.2.2.17311.43682).
[Ref3] Henry Garrett, “Connectivities of Neutrosophic Graphs in the terms of

Crisp Cycles”, ResearchGate 2022 (doi: 10.13140/RG.2.2.31917.77281). Three chapters are devised as “Weakest Path Connectivities”, “Strongest Path

Connectivities” and “Cycle Connectivities”.