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

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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.

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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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  1. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Junyao Pan & Xiuyun Guo

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  1. Junyao Pan
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  2. Xiuyun Guo
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Correspondence to Xiuyun Guo.

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The research of the work was partially supported by the National Natural Science Foundation of China (11771271).

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Pan, J., Guo, X. The Full Automorphism Groups, Determining Sets and Resolving Sets of Coprime Graphs. Graphs and Combinatorics 35, 485–501 (2019). https://doi.org/10.1007/s00373-019-02014-5

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  • DOI: https://doi.org/10.1007/s00373-019-02014-5

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