Periodicity of a graph

Periodicity of a graph --- периодичность графа.

Let [math]\displaystyle{ \phi }[/math] be a graph operator defined on the class [math]\displaystyle{ C_{f} }[/math] of all finite undirected graphs. For every positive integer [math]\displaystyle{ r }[/math] we define the power [math]\displaystyle{ \phi^{r} }[/math] so that [math]\displaystyle{ \phi^{1} = \phi }[/math] and for [math]\displaystyle{ r \geq 2 }[/math] the operator [math]\displaystyle{ \phi^{r} }[/math] is such that [math]\displaystyle{ \phi(\phi^{r-1}(G)) }[/math] for each [math]\displaystyle{ G \in C_{f} }[/math]. A graph [math]\displaystyle{ G \in C_{f} }[/math] is called [math]\displaystyle{ \phi }[/math]-periodic, if there exists a positive integer [math]\displaystyle{ r }[/math] such that [math]\displaystyle{ \phi^{r}(G) \cong G }[/math]. The minimum number [math]\displaystyle{ r }[/math] with this property is the periodicity of the graph [math]\displaystyle{ G }[/math] in the operator [math]\displaystyle{ \phi }[/math].