Notes on the Co-prime Order Graph of a Group

Authors

  • Shangjing Hao School of Science, Xi’an Shiyou University
  • Guo Zhong School of Information Science and Technology and Guangzhou Key laboratory of Multilingual Intelligent Processing, Guangdong University of Foreign Studies
  • Xuanlong Ma School of Science, Xi’an Shiyou University

DOI:

https://doi.org/10.7546/CRABS.2022.03.03

Keywords:

co-prime order graph, vertex-connectivity, group

Abstract

The co-prime order graph of a group $$G$$ is the graph with vertex set $$G$$, and two distinct elements $$x,y\in G$$ are adjacent if gcd$$(o(x),o(y))$$ is either $$1$$ or a prime, where $$o(x)$$ and $$o(y)$$ are the orders of $$x$$ and $$y$$, respectively. In this paper, we characterize finite groups whose co-prime order graphs are complete and classify finite groups whose co-prime order graphs are
planar, which generalizes some results by Banerjee [3]. We also compute the vertex-connectivity of the co-prime order graph of a cyclic group, a dihedral group and a generalized quaternion group, which answers a question by Banerjee [3].

Author Biographies

Shangjing Hao, School of Science, Xi’an Shiyou University

Mailing Address:
School of Science,
Xi’an Shiyou University
Xi’an 710065, P. R. China

E-mail: sjhao@163.com

Guo Zhong, School of Information Science and Technology and Guangzhou Key laboratory of Multilingual Intelligent Processing, Guangdong University of Foreign Studies

Mailing Address:
School of Information Science and Technology
Guangdong University of Foreign Studies
Guangzhou 510000, P. R. China

Guangzhou Key laboratory of Multilingual Intelligent Processing
Guangdong University of Foreign Studies
Guangzhou 510000, P. R. China

E-mail: guozhong@um.edu.mo

Xuanlong Ma, School of Science, Xi’an Shiyou University

Mailing Address:
School of Science,
Xi’an Shiyou University
Xi’an 710065, P. R. China

E-mail: xuanlma@xsyu.edu.cn

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Published

27-03-2022

How to Cite

[1]
S. . Hao, G. Zhong, and X. Ma, “Notes on the Co-prime Order Graph of a Group”, C. R. Acad. Bulg. Sci. , vol. 75, no. 3, pp. 340–348, Mar. 2022.

Issue

Section

Mathematics