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Graphs and Combinatorics

, Volume 35, Issue 2, pp 485–501 | Cite as

The Full Automorphism Groups, Determining Sets and Resolving Sets of Coprime Graphs

  • Junyao Pan
  • Xiuyun GuoEmail author
Original Paper
  • 19 Downloads

Abstract

The coprime graph is a graph \(TCG_n\) whose vertex set is \(\{1, 2, 3,\ldots ,n\}\), with two vertices i and j joined by an edge if and only if \(\hbox {gcd}(i, j)= 1\). In this paper we first determine the full automorphism group of the coprime graph, and then find the regularities for a set becoming a determining set or a resolving set in a coprime graph. Finally, we show that minimal determining sets of coprime graphs satisfy the exchange property and minimal resolving sets of coprime graphs do not satisfy the exchange property.

Keywords

Coprime graph Full automorphism groups Determining set Resolving set Exchange property 

Mathematics Subject Classification

05C25 05C12 20B05 20F65 

Notes

Acknowledgements

The authors would like to thank the referees for their valuable suggestions and useful comments contributed to the final version of this paper.

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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghaiPeople’s Republic of China

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