Graphs and Combinatorics

, Volume 35, Issue 2, pp 485–501

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

• Junyao Pan
• Xiuyun Guo
Original Paper

## 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.

## References

1. 1.
Ahlswede, R., Khachatrian, L.H.: On extremal sets without coprimes. Acta Arith. 66, 89–99 (1994)
2. 2.
Ahlswede, R., Khachatrian, L.H.: Maximal sets of numbers not containing k+1 pairwise coprime integers. Acta Arith. 72, 77–100 (1995)
3. 3.
Ahlswede, R., Khachatrian, L.H.: Sets of integers and quasi-integers with pairwise common divisors and a factor from a specified set of primes. Acta Arith. 75(3), 259–276 (1996)
4. 4.
Ahlswede, R., Blinovsky, V.: Maximal sets of numbers not containing k+1 pairwise coprimes and having divisors from a specified set of primes. J. Combin. Theory Ser. A 113, 1621–1628 (2008)
5. 5.
Albertson, M.O., Boutin, D.L.: Using determining sets to distinguish Kneser graphs. Electron. J. Combin. 14(1), 9 (2007) (Research Paper 20)Google Scholar
6. 6.
Albertson, M.O., Boutin, D.L.: Automorphisms and distinguishing numbers of geometric cliques. Discrete Comput. Geom. 39(4), 778–785 (2008)
7. 7.
Bailey, R.F., Cameron, P.J.: Base size, metric dimension and other invariants of groups and graphs. Bull. Lond. Math. Soc. 43(2), 209–242 (2011)
8. 8.
Biggs, N.: Algebraic Graph Theory, 2nd edn. Cambridge University Press, Cambridge (1993)
9. 9.
Boutin, D.L.: Identifying graph automorphisms using determining sets. Electron. J. Combin. 13(1), 12 (2006) (Research Paper 78)Google Scholar
10. 10.
Boutin, D.L.: Determining sets, resolving sets, and the exchange property. Graphs Combin. 25(6), 789–806 (2009)
11. 11.
Cameron, P.J., Fon-Der-Flaass, D.G.: Bases for permutation groups and matroids. Eur. J. Combin 16(6), 537–544 (1995)
12. 12.
Chartrand, G., Eroh, L., Johnson, M.A., Oellermann, O.R.: Resolvability in graphs and the metric dimension of a graph. Discrete Appl. Math 105, 99–113 (2000)
13. 13.
Dixon, J.D., Mortimer, B.: Permutation Groups, vol. 163, Graduate Texts in Mathematics. Springer, New York (1996)Google Scholar
14. 14.
Erdös, P.: Remarks in number theory, IV (in Hungarian). Mat. Lapok 13, 228–255 (1962)
15. 15.
Erdös, P., Sarkozy, G.N.: On cycles in the coprime graph of integers. Electron. J. Combin. 4(2), 11 (1997) (Research Paper 8, approx.)Google Scholar
16. 16.
Erwin, D., Harary, F.: Destroying automorphisms by fixing nodes. Discrete Math. 306, 3244–3252 (2006)
17. 17.
Graham, R.L., Gräotschel, M., Lovàsz, L. (eds.): Handbook of Combinatorics, vol. I, MIT Press, Cambridge (1995)Google Scholar
18. 18.
Meher, J., Murty, M.R.: Ramanujan’s proof of Bertrand’s postulate. Am. Math. Mon. 120(7), 650–653 (2013)
19. 19.
Pan, C., Pan, C.: Elementary number theory, 2th edn, Peking University Press, Beijing (2003) (Chinese)Google Scholar
20. 20.
Pomerance, C., Selfridge, J.L.: Proof of D. J. Newmans coprime mapping conjecture. Mathematika 27, 69–83 (1980)
21. 21.
Rao, S.N.: A creative review on coprime (prime) graphs. In: Lecture Notes, DST Work Shop (23–28 May 2011), WGTA, BHU, Varanasi, pp. 1–24 (2011)Google Scholar
22. 22.
Reinhard, D.: Graph Theory, 2nd edn. Springer, New York (1997)
23. 23.
Sander, J.W., Sander, T.: On the kernel of the coprime graph of integers. Integers 9, 569–579 (2009)
24. 24.
Slater, P.J.: Leaves of trees. Congr. Numer. 14, 549–559 (1975)
25. 25.
Tomescu, I., Imran, M.: R-sets and metric dimension of necklace graphs. Appl. Math. Inf. Sci 9(1), 63–67 (2015)

© Springer Japan KK, part of Springer Nature 2019

## Authors and Affiliations

• Junyao Pan
• 1
• Xiuyun Guo
• 1
1. 1.Department of MathematicsShanghai UniversityShanghaiPeople’s Republic of China