Preprint Article Version 7 Preserved in Portico This version is not peer-reviewed

Metric Dimension in fuzzy(neutrosophic) Graphs-VII

Version 1 : Received: 5 November 2021 / Approved: 8 November 2021 / Online: 8 November 2021 (13:21:27 CET)
Version 2 : Received: 8 November 2021 / Approved: 9 November 2021 / Online: 9 November 2021 (11:22:47 CET)
Version 3 : Received: 11 November 2021 / Approved: 12 November 2021 / Online: 12 November 2021 (14:46:47 CET)
Version 4 : Received: 17 November 2021 / Approved: 18 November 2021 / Online: 18 November 2021 (13:44:55 CET)
Version 5 : Received: 18 November 2021 / Approved: 19 November 2021 / Online: 19 November 2021 (14:51:27 CET)
Version 6 : Received: 19 November 2021 / Approved: 22 November 2021 / Online: 22 November 2021 (14:01:55 CET)
Version 7 : Received: 22 November 2021 / Approved: 26 November 2021 / Online: 26 November 2021 (10:03:20 CET)

How to cite: Garrett, H. Metric Dimension in fuzzy(neutrosophic) Graphs-VII. Preprints 2021, 2021110142 (doi: 10.20944/preprints202111.0142.v7). Garrett, H. Metric Dimension in fuzzy(neutrosophic) Graphs-VII. Preprints 2021, 2021110142 (doi: 10.20944/preprints202111.0142.v7).

Abstract

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.

Keywords

Fuzzy Graphs; Neutrosophic Graphs; Dimension

Subject

MATHEMATICS & COMPUTER SCIENCE, Applied Mathematics

Comments (1)

Comment 1
Received: 26 November 2021
Commenter: Henry Garrett
Commenter's Conflict of Interests: Author
Comment: It's necessary to note that this version is last version of this article, which is sent as "submit follow-up version"


The changes are as follow:

-Improved abstract
-Added contents for background
-Added one subsection for first section which is entitled "Motivation and Contributions"
-Added framework of article and motivation in "Motivation and Contributions
-Renamed section two from section one to "Preliminaries"
-Improved the first paragraph of  "Preliminaries"
-Changed slightly texts in last paragraph of  "Preliminaries"
-Changed some texts in the first paragraph of  "Definitions" in section 2.
-Added one section entitled "Applications" and added some texts in it.
-Added two subsections for section 7 and added some contents.
-Added section 8 entitled "Open Problems" and added some contents for it.
-Added one section entitled "Discussion" and it's section 9.
-Added section 10 entitled "Conclusion and Closing Remarks" and added some contents for it.
-Added one table concerning advantages and limitations of this article into section 10.
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