Geodesic
A note on periodicity of the 2-distance operator
Discussiones Mathematicae. Graph Theory, Tome 20 (2000) no. 2, pp. 267-269.

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The paper solves one problem by E. Prisner concerning the 2-distance operator T₂. This is an operator on the class C_f of all finite undirected graphs. If G is a graph from 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.
Keywords: 2-distance operator, complement of a graph 
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Zelinka, Bohdan. A note on periodicity of the 2-distance operator. Discussiones Mathematicae. Graph Theory, Tome 20 (2000) no. 2, pp. 267-269. http://geodesic.mathdoc.fr/item/DMGT_2000_20_2_a9/

[1] F. Harary, C. Hoede and D. Kadlacek, Graph-valued functions related to step graphs, J. Comb. Ing. Syst. Sci. 7 (1982) 231-246.

[2] E. Prisner, Graph Dynamics (Longman House, Burnt Mill, Harlow, 1995).