On the connectedness of some geometric graphs
Let \(S\subseteq R\). Then by \(G(S^ d)\) we denote a graph whose vertices are all the points of the real space \(R^ d\) with their coordinates in S and two vertices are connected by an edge if their distance is 1. In the paper it is proved that G(Q(\(\sqrt{k})^ d)\) is connected for all \(k\in N\), \(d\geq 5\); G(Q(\(\sqrt{k})^ 2)\) has countably infinitely many connected components for \(k=1,2,3,5,6(mod 8)\), and that for every algebraic number a, \(G(Q(a)^ d)\) is a connected graph for every large d.
scientific article; zbMATH DE number 4202296