Metric Dimension in Fuzzy Graphs and Neutrosophic Graphs

In this study, the term dimension is introduced on fuzzy graphs and neutrosophic graphs. The classes of these specific graphs are chosen to obtain some results based on dimension. The types of crisp notions and fuzzy 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.


30
Example 2.4. Let G = {G 1 , G 2 , G 3 } be a collection of fuzzy graphs with common 31 fuzzy vertex set and a subset of fuzzy edge set as figure (2). By applying Table (2), the 32 1-set is explored which its cardinality is minimum. {f 2 } and {f 4 } are 1-set which has 33 minimum cardinality amid all sets of vertices. {f 4 } is as fuzzy resolving set as {f 6 } is.

34
Thus there's no fuzzy metric set but {f 4 } and {f 6 }. f 6 as fuzzy-resolves all given couple 35 of vertices as f 4 . Therefore one is fuzzy metric number of G and 0.13 is fuzzy metric 36 dimension of G. By using Table (2), f 4 fuzzy-resolves all given couple of vertices.  Proof. Let l be a leaf. For every given a couple of vertices f i and f j , we get Since if we reassign indexes to vertices such that every vertex f i and l have i vertices amid themselves, then d(l,   Then fuzzy metric number is n − 2.

64
Corollary 3.8. Let G be a fuzzy graph. The number of fuzzy twin vertices is n − 1.

65
Then G is fixed-edge fuzzy graph. 66

3/6
Corollary 3.9. Let G be a fixed-vertex fuzzy graph. The number of fuzzy twin vertices 67 is n − 1. Then fuzzy metric number is n − 2, fuzzy metric dimension is (n − 2)σ(m) 68 where m is fuzzy twin vertex with a vertex. Every (n − 2)-set including fuzzy twin 69 vertices is fuzzy metric set.

70
Proposition 3.10. Let G be a fixed-vertex fuzzy graph such that it's fuzzy complete.

71
Then fuzzy metric number is n − 1, fuzzy metric dimension is (n − 1)σ(m) where m is a 72 given vertex. Every (n − 1)-set is fuzzy metric set.   s, t) where G is 102 a given fuzzy graph. It means that t and t aren't resolved in any given fuzzy graph. t 103 and t are arbitrary so fuzzy twin vertices aren't resolved in any given fuzzy graph.
Theorem 3.16. Let G be a family of fuzzy graphs with common vertex set and G ∈ G 111 is a fixed-vertex fuzzy graph such that it's fuzzy complete. Then simultaneously fuzzy 112 metric number is n − 1, simultaneously fuzzy metric dimension is (n − 1)σ(m) where m 113 is a given vertex. Every (n − 1)-set is simultaneously fuzzy metric set for G.

114
Proof. G is fixed-vertex fuzzy graph and it's fuzzy complete. So by Proposition (3.15), 115 we get every couple of vertices in fuzzy complete are fuzzy twin vertices. So every 116 couple of vertices, by Theorem (3.14), aren't resolved.

117
Theorem 3.17. Let G be a family of fuzzy graphs with common vertex set and for 118 every given couple of vertices, there's a G ∈ G such that in that, they're fuzzy twin 119 vertices. Then simultaneously fuzzy metric number is n − 1, simultaneously fuzzy metric 120 dimension is (n − 1)σ(m) where m is a given vertex. Every (n − 1)-set is simultaneously 121 fuzzy metric set for G.

122
Theorem 3.18. Let G be a family of fuzzy graphs with common vertex set. If G 123 contains three fixed-vertex fuzzy stars with different center, then simultaneously fuzzy 124 metric number is n − 2, simultaneously fuzzy metric dimension is (n − 2)σ(m) where m 125 is a given vertex. Every (n − 2)-set is simultaneously fuzzy metric set for G.