Metric Dimension in fuzzy(neutrosophic) Graphs-VII

New notion of dimension as set, as two optimal numbers including metric number, dimension number and as optimal set are introduced in individual framework and in formation of family. Behaviors of twin and antipodal are explored in fuzzy(neutrosophic) graphs. Fuzzy(neutrosophic) graphs, under conditions, fixed-edges, fixed-vertex and strong fixed-vertex are studied. Some classes as path, cycle, complete, strong, t-partite, bipartite, star and wheel in the formation of individual case and in the case, they form a family are studied in the term of dimension. Fuzzification (neutrosofication) of twin vertices but using crisp concept of antipodal vertices are another approaches of this study. Thus defining two notions concerning vertices which one of them is fuzzy(neutrosophic) titled twin and another is crisp titled antipodal to study the behaviors of cycles which are partitioned into even and odd, are concluded. Classes of cycles according to antipodal vertices are divided into two classes as even and odd. Parity of the number of edges in cycle causes to have two subsections under the section is entitled to antipodal vertices. In this study, the term dimension is introduced on fuzzy(neutrosophic) graphs. The locations of objects by a set of some junctions which have distinct distance from any couple of objects out of the set, are determined. Thus it’s possible to have the locations of objects outside of this set by assigning partial number to any objects. The classes of these specific graphs are chosen to obtain some results based on dimension. The types of crisp notions and fuzzy(neutrosophic) notions are used to make sense about the material of this study and the outline of this study uses some new notions which are crisp and fuzzy(neutrosophic). Some questions and problems are posed concerning ways to do further studies on this topic. Basic familiarities with fuzzy(neutrosophic) graph theory and graph theory are proposed for this article.

given. In section (4), one idea about specific fuzzy(neutrosophic) vertices is used to 23 form the results for fuzzy(neutrosophic) graphs and family of them. In section (5), one 24 idea about specific crisp vertices is used to form the results for fuzzy(neutrosophic) 25 graphs and family of them especially fuzzy(neutrosophic) cycles as two subsections, In 26 section (6), the results are extended for fuzzy(neutrosophic) graphs and family of them. 27 In section (7), two applications are posed for fuzzy(neutrosophic) graphs and family of 28 them. In section (8), some problems and questions for further studies are proposed. In 29 section (9), gentle discussion about results and applications are featured. In section 30 (10), a brief overview concerning advantages and limitations of this study alongside 31 conclusions are formed. 32

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To clarify about the models, I use some definitions and results, and in this way, results 34 have a key role to make sense about the definitions and to introduce new ways to use on 35 these models in the terms of new notions. For instance, the concept of complete is used 36 to specialize a graph in every environment. To differentiate, I use an adjective or prefix 37 in every definition. Two adjectives "fuzzy" and "neutrosophic" are used to distinguish 38 every graph or classes of graph or any notion on them. complete where ∀x ∈ V, ∀y ∈ V, xy ∈ E. A fuzzy graph G : (σ, µ) is called fuzzy 46 complete where it's complete and µ(xy) = σ(x) ∧ σ(y) for all xy ∈ E. A neutrosophic 47 graph G : (σ, µ) is called a neutrosophic complete where it's complete and crisp graph G : (V, E) is called crisp path with length n from v 0 to v n where 55 v i v i+1 ∈ E, i = 0, 1, · · · , n − 1. If one edge is incident to a vertex, the vertex is called 56 leaf. A path v 0 , v 1 , · · · , v n is called fuzzy path where 57 µ(v i v i+1 ) > 0, i = 0, 1, · · · , n − 1. A path v 0 , v 1 , · · · , v n is called neutrosophic path 58 where µ(v i v i+1 ) > 0, i = 0, 1, · · · , n − 1. Let P : v 0 , v 1 , · · · , v n be fuzzy(neutrosophic) 59 path from v 0 to v n such that it has minimum number of vertices as possible, then cycle v 0 , v 1 , · · · , v 0 is called fuzzy cycle where there are two edges xy and uv such that 63 cycle where there are two edges xy and uv such that where M is fuzzy(neutrosophic)-resolving set, then M is called 97 fuzzy(neutrosophic)-metric set of G.
Vertices    Table 4. Distances of Vertices from set of vertices {f 6 } in Family of fuzzy(neutrosophic) Graphs G.
Proof. Let l be a leaf. For every given a couple of vertices f i and f j , I get d(l, f i ) = d(l, f j ). Since if I reassign indexes to vertices such that every vertex f i and l have i vertices amid themselves, then d(l, 138 Proof. Let f and f be a couple of vertices. For every given a couple of vertices f i and 139  Proof. By fuzzy(neutrosophic) graphs with fuzzy(neutrosophic) common vertex set, G 245 is fixed-vertex fuzzy(neutrosophic) graph. It's fuzzy(neutrosophic) complete. So by 246 Theorem (4.11), I get intended result.

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Theorem 4.13. Let G be a family of fuzzy(neutrosophic) graphs with common vertex 248 set and for every given couple of vertices, there's a G ∈ G such that in that, they're vertex, there's no vertex such that they're antipodal vertices.

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Proof. Let G be a fixed-edge odd fuzzy(neutrosophic) cycle. if x is a given vertex. Then 362 there are two vertices u and v such that d(x, u) = d(x, v) = D(G). It implies they aren't 363 antipodal vertices.
this way, some crisp notions like antipodal vertices are defined to use as a tool to study 502 fuzzy(neutrosophic) cycles as individual and as family. Also, some fuzzy(neutrosophic) 503 notions like fuzzy(neutrosophic) twin vertices are defined to use as a tool to study 504 general classes of fuzzy(neutrosophic) graphs as individual and as family. Mixed family 505 of fuzzy(neutrosophic) graphs are slightly studied by using fuzzy(neutrosophic) twin 506 vertices and other ideas as individual and as family. In Table ( 5), I mention some 507 advantages and limitations concerning this article and its proposed notions.