Is it necessary that the powers of regular graphs regular?

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Aurea Rosal

Sir, by power you mean cartesian product of a graph by itself?

Sudev Naduvath

The k-th power of a graph G is the graph obtained by joining every pair vertices in G which are at a distance at most k. The first figure is the square of the cycle C7 and the second figure is the cube of the cycle C8.

Hamdy Mohamed Hafez

No, not necessary, consider the graph in fig.1 in the following paper

http://arxiv.org/abs/1012.2950 . the graph is regular while its square is not.