0032 | Beyond Neutrosophic Graphs (E-Publisher)

Henry Garrett, (2022). “Beyond Neutrosophic Graphs”, Ohio: E-publishing: 873 Educational Publisher 1091 West 1st Ave Grandview Heights, Ohio 43212 874 United States. ISBN: 978-1-59973-725-6 875

(http://fs.unm.edu/BeyondNeutrosophicGraphs.pdf).

In this outlet, a journey amid three models are designed. Graphs, fuzzy graphs and neutrosophic graphs are three models which form main parts. Assigning one specific number with some conditions to vertices and edges of graphs make them to be titled as fuzzy graphs and assigning three specific numbers with some conditions to vertices and edges of graphs make them to be titled as neutrosophic graphs. In other viewpoint, neutrosophic graphs are 3-array fuzzy graphs which every things are triple. To make more sense, the well-known graphs are defined in new ways. For example, crisp complete, fuzzy complete and neutrosophic complete when the context is about being complete in every model. New notions are defined in the comparable structures on these three models to understand the behaviors of these models according to the notions. Different edges define new form of connections amid vertices. Thus defining new notion of coloring is possible when the connections of vertices which determine new color and it’s decider whether using new color or not, have been considered if they’ve special edges. The tools to define specific edges are studied. One notion is to use the connectedness to have two different types of numbers which are neutrosophic chromatic number and chromatic number. Other notion is to use the idea of neutrosophic strong to get specific edges which are eligible to define new numbers. Some classes of neutrosophic graphs are studied in the the terms of different types of chromatic numbers and neutrosophic chromatic numbers. This book is based on neutrosophic graph theory which is designed to study different types of coloring in that graphs to get new ideas and new results. The results concern specific classes of neutrosophic graphs. In this book, idea of neutrosophic is applied into the setting of hypergraphs and n-SuperHyperGraphs. New setting has the name neutrosophic hypergraphs and neutrosophic n-SuperHyperGraphs. Also, idea of close numbers and super-close numbers are applied to study. The idea of closing numbers and super-closing numbers are some names for (dual) super-coloring and (dual) super-resolving alongside (dual) super-dominating which give us a set and number arising from hyper-vertices and super-vertices alongside their relations in neutrosophic hypergraphs and neutrosophic n- SuperHyperGraphs. When hyper-vertices and super-vertices are too close, idea of (dual) super-coloring and (dual) super-resolving alongside (dual) super- dominating are introduced to study the behaviors of too close hyper-vertices and super-vertices. In this book, idea of neutrosophic is applied into the setting of hypergraphs and n-SuperHyperGraphs. New setting has the name neutrosophic hypergraphs and neutrosophic n-SuperHyperGraphs. Also, idea of close numbers and super-close numbers are applied to study. The idea of closing numbers

i

Abstract

and super-closing numbers are some names for (dual) super-coloring and (dual) super-resolving alongside (dual) super-dominating which give us a set and number arising from hyper-vertices and super-vertices alongside their relations in neutrosophic hypergraphs and neutrosophic n-SuperHyperGraphs. When hyper-vertices and super-vertices are too close, idea of (dual) super-coloring and (dual) super-resolving alongside (dual) super-dominating are introduced to study the behaviors of too close hyper-vertices and super-vertices. New setting is introduced to study the alliances. Alliances are about a set of vertices which are applied into the setting of neutrosophic graphs. Neighborhood has the key role to define these notions. Also, neighborhood is defined based on the edges, strong edges and some edges which are coming from connectedness. These three types of edges get a framework as neighborhood and after that, too close vertices have key role to define offensive alliance, defensive alliance, t-offensive alliance, and t- defensive alliance based on three types of edges, common edges, strong edges and some edges which are coming from connectedness. The structure of set is studied and general results are obtained. Also, some classes of neutrosophic graphs containing complete, empty, path, cycle, bipartite, t-partite, star and wheel are investigated in the terms of set, minimal set, number, and neutrosophic number. In this study, there’s an open way to extend these results into the family of these classes of neutrosophic graphs. The family of neutrosophic graphs aren’t study but it seems that analogous results are determined. There’s a question. How can be related to each other, two sets partitioning the vertex set of a graph? The ideas of neighborhood and neighbors based on different edges illustrate open way to get results. A set is alliance when two sets partitioning vertex set have uniform structure. All members of set have different amount of neighbors in the set and out of set. It leads us to the notion of offensive and defensive. New ideas, offensive alliance, defensive alliance, t-offensive alliance, t-defensive alliance, strong offensive alliance, strong defensive alliance, strong t-offensive alliance, strong t-defensive alliance, connected offensive alliance, connected defensive alliance, connected t-offensive alliance, and connected t-defensive alliance are introduced. Two numbers concerning cardinality and neutrosophic cardinality of alliances are introduced. A set is alliance when its complement make a relation in the terms of neighborhood. Different edges make different neighborhoods. Three types of edges are applied to define three styles of neighborhoods. General edges, strong edges and connected edges are used where connected edges are the edges arising from connectedness amid two endpoints of the edges. These notions are applied into neutrosophic graphs as individuals and family of them. Independent set as an alliance is a special set which has no neighbor inside and it implies some drawbacks for this notions. Finding special sets which are well-known, is an open way to purse this study. Special set which its members have only one neighbor inside, characterize the connected components where the cardinality of its complement is the number of connected components. Some problems are proposed to pursue this study. Basic familiarities with graph theory and neutrosophic graph theory are proposed for this book. New setting is introduced to study the global offensive alliance. Global offensive alliance is about a set of vertices which are applied into the setting of neutrosophic graphs. Neighborhood has the key role to define this notion. Also, neighborhood is defined based on strong edges. Strong edge gets a framework as neighborhood and after that, too close vertices have key role to define global offensive alliance based on strong edges. The structure of set is studied and general results are

ii

obtained. Also, some classes of neutrosophic graphs containing complete, empty, path, cycle, star, and wheel are investigated in the terms of set, minimal set, number, and neutrosophic number. Neutrosophic number is defined in new way. It’s first time to define this type of neutrosophic number in the way that, three values of a vertex are used and they’ve same share to construct this number. It’s called “modified neutrosophic number”. Summation of three values of vertex makes one number and applying it to a set makes neutrosophic number of set. This approach facilitates identifying minimal set and optimal set which forms minimal-global-offensive-alliance number and minimal-global-offensive-alliance- neutrosophic number. Two different types of sets namely global-offensive alliance and minimal-global-offensive alliance are defined. Global-offensive alliance identifies the sets in general vision but minimal-global-offensive alliance takes focus on the sets which deleting a vertex is impossible. Minimal-global- offensive-alliance number is about minimum cardinality amid the cardinalities of all minimal-global-offensive alliances in a given neutrosophic graph. New notions are applied in the settings both individual and family. Family of neutrosophic graphs is studied in the way that, the family only contains same classes of neutrosophic graphs. Three types of family of neutrosophic graphs including m-family of neutrosophic stars with common neutrosophic vertex set, m-family of odd complete graphs with common neutrosophic vertex set, and m-family of odd complete graphs with common neutrosophic vertex set are studied. The results are about minimal-global-offensive alliance, minimal-global- offensive-alliance number and its corresponded sets, minimal-global-offensive- alliance-neutrosophic number and its corresponded sets, and characterizing all minimal-global-offensive alliances. The connection of global-offensive-alliances with dominating set and chromatic number are obtained. The number of connected components has some relations with this new concept and it gets some results. Some classes of neutrosophic graphs behave differently when the parity of vertices are different and in this case, path, cycle, and complete illustrate these behaviors. Two applications concerning complete model as individual and family, under the titles of time table and scheduling conclude the results and they give more clarifications. In this study, there’s an open way to extend these results into the family of these classes of neutrosophic graphs. The family of neutrosophic graphs aren’t study deeply and with more results but it seems that analogous results are determined. Slight progress is obtained in the family of these models but there are open avenues to study family of other models as same models and different models. There’s a question. How can be related to each other, two sets partitioning the vertex set of a graph? The ideas of neighborhood and neighbors based on strong edges illustrate open way to get results. A set is global offensive alliance when two sets partitioning vertex set have uniform structure. All members of set have more amount of neighbors in the set than out of set. It leads us to the notion of global offensive alliance. Different edges make different neighborhoods but it’s used one style edge titled strong edge. These notions are applied into neutrosophic graphs as individuals and family of them. Independent set as an alliance is a special set which has no neighbor inside and it implies some drawbacks for these notions. Finding special sets which are well-known, is an open way to purse this study. Special set which its members have only one neighbor inside, characterize the connected components where the cardinality of its complement is the number of connected components. Some problems are proposed to pursue this study. Basic

iii

Abstract

familiarities with graph theory and neutrosophic graph theory are proposed for this book.

New notions are defined in the comparable structures on these three models to understand the behaviors of these models according to the notions. This book is based on neutrosophic graph theory which is designed to study different types of coloring in that graphs to get new ideas and new results. The results concern specific classes of neutrosophic graphs. New notions are defined in the comparable structures on these three models to understand the behaviors of these models according to the notions. This book is based on neutrosophic graph theory which is designed to study different types of coloring in that graphs to get new ideas and new results. The results concern specific classes of neutrosophic graphs.