On connected components and perfect codes of proper order graphs of finite groups (Q6537363)

The order graph of the group \(G\) is the graph whose vertices are the elements of \(G\), with adjacency between vertices if the order of one of the corresponding group elements is a divisor of the order of the other. Clearly, the vertex corresponding to the identity is adjacent to every other vertex of this graph, so in this paper, the authors define the proper order graph as the induced subgraph on the vertices corresponding to the non-identity elements of the group.\N\NThe authors determine the connected components of any such graph. They also study the perfect codes (also known as efficient dominating sets) for these graphs, determining them for various classes of groups including nilpotent groups. The paper is fairly short and easy to read.