Open AccessNotes on the Co-prime Order Graph of a Group
Author(s) -
Shangjing Hao,
Guo Zhang,
Xuanlong Ma
Publication year - 2022
Publication title -
dokladi na bʺlgarskata akademiâ na naukite
Language(s) - English
Resource type - Journals
eISSN - 2367-5535
pISSN - 1310-1331
DOI - 10.7546/crabs.2022.03.03
Subject(s) - combinatorics , dihedral group , mathematics , finite group , prime (order theory) , vertex (graph theory) , cyclic group , graph , discrete mathematics , order (exchange) , group (periodic table) , physics , abelian group , finance , quantum mechanics , economics
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].