The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes
Joint Authors
Khemmani, Varanoot
Isariyapalakul, Supachoke
Source
International Journal of Mathematics and Mathematical Sciences
Issue
Vol. 2018, Issue 2018 (31 Dec. 2018), pp.1-6, 6 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2018-08-01
Country of Publication
Egypt
No. of Pages
6
Main Subjects
Abstract EN
For a set W = w 1 , w 2 , … , w k of vertices and a vertex v of a connected graph G , the multirepresentation of v with respect to W is the k -multiset m r ( v ∣ W ) = d v , w 1 , d v , w 2 , … , d v , w k , where d ( v , w i ) is the distance between the vertices v and w i for i = 1,2 , … , k .
The set W is a multiresolving set of G if every two distinct vertices of G have distinct multirepresentations with respect to W .
The minimum cardinality of a multiresolving set of G is the multidimension d i m M ( G ) of G .
It is shown that, for every pair k , n of integers with k ≥ 3 and n ≥ 3 ( k - 1 ) , there is a connected graph G of order n with d i m M ( G ) = k .
For a multiset { a 1 , a 2 , … , a k } and an integer c , we define { a 1 , a 2 , … , a k } + c , c , … , c = a 1 + c , a 2 + c , … , a k + c .
A multisimilar equivalence relation R W on V ( G ) with respect to W is defined by u R W v if m r ( u ∣ W ) = m r v ∣ W + c W u , v , c W u , v , … , c W u , v for some integer c W ( u , v ) .
We study the relationship between the elements in multirepresentations of vertices that belong to the same multisimilar equivalence class and also establish the upper bound for the cardinality of a multisimilar equivalence class.
Moreover, a multiresolving set with prescribed multisimilar equivalence classes is presented.
American Psychological Association (APA)
Khemmani, Varanoot& Isariyapalakul, Supachoke. 2018. The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes. International Journal of Mathematics and Mathematical Sciences،Vol. 2018, no. 2018, pp.1-6.
https://search.emarefa.net/detail/BIM-1173530
Modern Language Association (MLA)
Khemmani, Varanoot& Isariyapalakul, Supachoke. The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes. International Journal of Mathematics and Mathematical Sciences No. 2018 (2018), pp.1-6.
https://search.emarefa.net/detail/BIM-1173530
American Medical Association (AMA)
Khemmani, Varanoot& Isariyapalakul, Supachoke. The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes. International Journal of Mathematics and Mathematical Sciences. 2018. Vol. 2018, no. 2018, pp.1-6.
https://search.emarefa.net/detail/BIM-1173530
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-1173530