A note on periodicity of the 2-distance operator

The paper solves one problem by E. Prisner concerning the 2-distance operator T₂. This is an operator on the class ${\displaystyle C_f}$ of all finite undirected graphs. If G is a graph from ${\displaystyle C_f}$, then T₂(G) is the graph with the same vertex set as G in which two vertices are adjacent if and only if their distance in G is 2. E. Prisner asks whether the periodicity ≥ 3 is possible for T₂. In this paper an affirmative answer is given. A result concerning the periodicity 2 is added.