Global Powerful Alliance in Strong Neutrosophic Graphs

New setting is introduced to study the global powerful alliance. Global powerful 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 powerful alliance based on strong edges. The structure of set is studied and general results are obtained. Also, some classes of neutrosophic graphs excluding empty, path, star, and wheel and containing complete, cycle and r-regular-strong are investigated in the terms of set, minimal set, number, and neutrosophic number. Neutrosophic number is used in this way. It’s applied to use the 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-powerful-alliance number and minimal-global-powerful-alliance-neutrosophic number. Two different types of sets namely global-powerful alliance and minimal-global-powerful alliance are defined. Global-powerful alliance identifies the sets in general vision but minimal-global-powerful alliance takes focus on the sets which deleting a vertex is impossible. Minimal-global-powerful-alliance number is about minimum cardinality amid the cardinalities of all minimal-global-powerful alliances in a given neutrosophic graph. New notions are applied in the settings both individual and family. Family of neutrosophic graphs has an open avenue, in the way that, the family only contains same classes of neutrosophic graphs. The results are about minimal-global-powerful alliance, minimal-global-powerful-alliance number and its corresponded sets, minimal-global-powerful-alliance-neutrosophic number and its corresponded sets, and characterizing all minimal-global-powerful alliances, minimal-t-powerful alliance, minimal-t-powerful-alliance number and its corresponded sets, minimal-t-powerful-alliance-neutrosophic number and its corresponded sets, and characterizing all minimal-t-powerful alliances. The connections amid t-powerful-alliances 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, 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 and closing remarks. 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 powerful 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 and reversely for non-members of set with less members in the way that the set is simultaneously t-offensive and (t-2)-defensive. A set is global if t=0. It leads us to the notion of global powerful 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 familiarities with graph theory and neutrosophic graph theory are proposed for this article. Refs. [2, 14,15], graphs and new notions on them in Refs. [5][6][7][8][9][10][11][12], neutrosophic graphs 3 in Ref. [3], studies on neutrosophic graphs in Ref. [1], relevant definitions of other 4 graphs based on fuzzy graphs in Ref. [13], related definitions of other graphs based on 5 neutrosophic graphs in Ref. [4], are proposed. 6 In this section, I use two subsections to illustrate a perspective about the 7 background of this study. In this study, there's an idea which could be considered as a motivation. 10 minimal-global-powerful-alliance-neutrosophic number are the cases of study in the 24 settings of individuals and in settings of families. Also, there are some avenues to 25 extend these notions. 26 The framework of this study is as follows. In the beginning, I introduce basic 27 definitions to clarify about preliminaries. In subsection "Preliminaries", new notions of 28 global-powerful alliance, minimal-global-powerful alliance, 29 minimal-global-powerful-alliance number, and   1]. We add one condition on it and we use special case of neutrosophic graph but with same name. The added condition is as follows, for every x i x i+1 ∈ E, i = 0, 1, · · · , n − 1; (iii) : connectedness amid vertices x 0 and x n is (iv) : a sequence of vertices P : x 0 , x 1 , · · · , x n is called cycle where 75 x i x i+1 ∈ E, i = 0, 1, · · · , n − 1 and there are two edges xy and uv such that New notion is defined between two types of neighborhoods for a fixed vertex. A |S| and it's denoted by Γ; and it's denoted by Γ s .

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In the next result, the notions of t-defensive alliance and t-offensive alliance have 105 been extended to present the classes of defensive alliance and offensive alliance which 106 hold when one type of them holds for a given set of vertices. (ii) if s ≤ t and a set S of vertices is t-offensive alliance, then S is s-offensive alliance. 112 Proof. (i). Suppose N T G : (V, E, σ, µ) is a strong neutrosophic graph. Consider a set S 113 of vertices is t-defensive alliance. Then Thus S is s-defensive alliance.

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(ii). Suppose N T G : (V, E, σ, µ) is a strong neutrosophic graph. Consider a set S of 119 vertices is t-offensive alliance. Then Thus S is s-offensive alliance.

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As a consequence of previous result, the relations amid a set which is both t-offensive 125 alliance and t-defensive alliance lead us toward the notion of t-powerful alliance. Proposition 1.8. Let N T G : (V, E, σ, µ) be a strong neutrosophic graph. Then 127 following statements hold; 128 (i) if s ≥ t + 2 and a set S of vertices is t-defensive alliance, then S is s-powerful 129 alliance;

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(ii) if s ≤ t and a set S of vertices is t-offensive alliance, then S is t-powerful alliance. 131 Proof. (i). Suppose N T G : (V, E, σ, µ) is a strong neutrosophic graph. Consider a set S 132 of vertices is t-defensive alliance. Then Thus S is (s-2)-defensive alliance. By S is (s-2)-defensive alliance and S is s-offensive 144 alliance, S is s-powerful alliance.
Thus S is 2-offensive alliance.
Thus S is r-defensive alliance.

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(iv). Suppose N T G : (V, E, σ, µ) is a r-regular-strong-neutrosophic graph. Then Thus S is r-offensive alliance. Then following statements hold; is a r-regular-strong-neutrosophic graph and 188 2-defensive alliance. Then is a r-regular-strong-neutrosophic graph and 194 2-offensive alliance. Then is a r-regular-strong-neutrosophic graph and 200 r-defensive alliance.
is a r-regular-strong-neutrosophic graph and 206 r-offensive alliance. Then which is complete. Then following statements hold; Proof. (i). Suppose N T G : (V, E, σ, µ) is a r-regular-strong-neutrosophic graph and 224 2-defensive alliance. Then is a r-regular-strong-neutrosophic graph and 230 2-offensive alliance. Then As a special case of r-regular, complete is an attribute. This property facilitates the 249 results when the condition is about the neighbors inside fixed set to determine Thus S is 2-defensive alliance.

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As a special case, cycle neutrosophic graph gets specific result excerpt from 2-regular 307 neutrosophic graph. 2-defensive alliance and 2-offensive alliance get some results about 308 the neighbors inside fixed set which their number is at most 2. Also, 2-defensive alliance 309 and 2-offensive alliance get some results about the neighbors outside of fixed set which 310 their number is at most 2.
Thus S is 2-offensive alliance.

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Designing the programs to achieve some goals is general approach to apply on some 425 issues to function properly. Separation has key role in the context of this style.

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Separating the duration of work which are consecutive, is the matter and it has 427 importantance to avoid mixing up.  Step 2. (Issue) Scheduling of program has faced with difficulties to differ amid 432 consecutive section. Beyond that, sometimes sections are not the same.

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Step 3. (Model) The situation is designed as a model. The model uses data to assign 434 every section and to assign to relation amid section, three numbers belong unit 435 interval to state indeterminacy, possibilities and determinacy. There's one 436 restriction in that, the numbers amid two sections are at least the number of the 437 relation amid them. Step 4. (Solution) The neutrosophic graph and its global offensive alliance as model, 441 propose to use specific set. Every subject has connection with every given subject. 442 Thus the connection is applied as possible and the model demonstrates full In this section, some questions and problems are proposed to give some avenues to 533 pursue this study. The structures of the definitions and results give some ideas to make 534 new settings which are eligible to extend and to create new study.

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Notion concerning alliance is defined in neutrosophic graphs. Neutrosophic number 536 is also introduced. Thus,